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Chapter 4 - Entanglement

2011-11-07T17:13:55Z

Stempel2:

===Introduction===

Quantum entanglement is the most uniquely quantum mechanical property of quantum systems. It is also believed to be responsible for the advantages that quantum computing systems have over classical computing systems. Entangled states puzzled the founders of quantum theory, including Einstein. For these reasons, it is not surprising that they have become a central part of many investigations into quantum theory and especially quantum information theory.

There are many open problems in this area of research. Some of the most basic and fundamental questions about entangled states are still unanswered. For example, given a mixed-state density matrix for a quantum system, with few exceptions we still do not know how to tell if the systems being described by that matrix are entangled or not. Also, although there are ways with which to quantify the entanglement in a system of particles, these quantities are notoriously difficult to calculate. Here again, with few exceptions, we do not know how to calculate the amount of entanglement in a system analytically.

What we do understand and what we can explain is the entanglement between bipartite systems that are describable by pure quantum states. In this chapter, we will first talk about the problem of entanglement in quantum mechanics in general. We will then move on to its relevancy in quantum computing, starting with pure states of qubits. Extensions and generalizations will be discussed in later sections.

===EPR Paradox===

Before diving into entanglement in the context of quantum computing and information, it would be prudent to discuss the now-famous EPR paradox. It was first proposed in a paper by Einstein, Polosky, and Rosen in 1935. The original paper does include a fairly simple mathematical explanation of the paradox---it is, however, not really necessary as the thought experiment is quite easily understood conceptually with (mostly) words. A slightly simplified version of the experiment will be given here.

Suppose a neutral pi meson, which has no spin, is at rest. It then decays into an electron and a positron, necessarily going in opposite directions. The wave function can now be written as

{{Equation|<math>
\left\vert \Psi_-\right\rangle = \frac{1}{\sqrt{2}}(\left\vert \uparrow\downarrow \right\rangle - \left\vert \downarrow\uparrow \right\rangle). </math>|4.1}}

As can be seen, we now have a system of two particles that have a correlated spin---one being up and the other being down---with an equal probability for each configuration being the outcome of a measurement. The system is said to be ''entangled'', as a measurement on one will guarantee that the other particle is in the correlated state. In other words, it cannot be written as <math>\left\vert \psi_1\right\rangle \left\vert \psi_2\right\rangle </math>.

Now what is the significance here? It all depends on what interpretation of quantum mechanics is being used. The orthodox position says that the wave function is the complete representation of the system. When the measurement occurs, the wave function collapses, changing the system.

But, in the context of EPR, how can this be? Imagine the entangled electron-positron pair are at opposite ends of the galaxy when one of them is measured. The conservation of angular momentum says that the other particle all of the way on the other side of the galaxy must ''instantly'' be the opposite spin as the measured particle. EPR argued that this is a violation of locality, which says an effect cannot travel faster than the speed of light. If the very action of measurement on one particle is what caused the other particle to realize the opposite spin, then locality has been violated. Therefore, the measurement could not have caused the collapse of the wave function.

EPR concluded that this proves that quantum mechanics is incomplete---that the wave function is missing some information. There was no "spooky-action-at-a-distance," there must be some underlying property that is absent from the wave function. Einstein rejected the notion that a measurement caused this quasi-mystical collapse of the wave function---the particles do not care if they are being watched or not.

===Bell's Theorem===

The peculiarities of the EPR paradox were convincing enough to drive many to examine possible "hidden variable theories." The basic idea is that there exists a quantity, often denoted by <math>\lambda \,\!</math>, that must be included in the wave function to completely describe the system. J.S. Bell very elegantly showed in 1964 that this is not the case, using the very thought experiment (although slightly modified) that EPR proposed.

Suppose we have another pion at rest about to decay with detectors oriented equidistant and on opposite sides, ready to measure the spin of the electron and positron. Further suppose that, unlike the previous scenario, these detectors can be rotated in order to detect the spin in the direction of unit vectors <math>\vec{a}\,\!</math> and <math>\vec{b}</math> for the electron and positron respectively.

When the electron and positron pair strikes the detectors, a spin up (<math>+1\,\!</math>) or spin down (<math>-1\,\!</math>) is registered. The product of the results is then examined. If they are oriented parallel, where <math> \vec{a} = \vec{b}\,\!</math>, then the result will be -1. If anti-parallel, the result is then +1. The averages are, obviously,

{{Equation|<math>\begin{align}
P(\vec{a}, \vec{a}) &= -1 , \\
P(\vec{a}, -\vec{a}) &= +1 .
\end{align}\,\!</math>|4.2}}

Quantum mechanics tells us that for arbitrary vectors,

{{Equation|<math> P(\vec{a}, \vec{b}) = -\vec{a} \cdot \vec{b}. \,\!</math>|4.3}}

We can now introduce the hidden variable, <math>\lambda\,\!</math>. This can represent ''any'' possible amount of variables that complete the description of the system and allow for locality. We then define some functions, <math>A(\vec{a},\lambda)</math> and <math>B(\vec{b},\lambda)\,\!,</math> that will give the results for the measurement (either +1 or -1) for the electron and positron respectively.

The locality assumption tells us that the orientation of one detector will not affect the outcome of the measurement of the other detector; one can imagine a scenario where the orientation is chosen at a time too late for any information to be transferred slower than light. It must also be true that, when the detectors are parallel, the results must be

{{Equation|<math> A(\vec{a},\lambda) = -B(\vec{a},\lambda) \,\!</math>|4.4}}

due to the conservation of angular momentum. Let us also define a probability density, <math>\rho (\lambda),\,\!</math> for the hidden variable. Since we know nothing of <math>\lambda\,\!</math>, this can be ''anything'' as long as it is non-negative and normalizable (<math>\int \rho (\lambda)d(\lambda) = 1\,\!</math>). We can now look at the product of the measurements,

{{Equation|<math>P(\vec{a},\vec{b}) = \int \rho(\lambda)A(\vec{a},\lambda)B(\vec{b},\lambda)d\lambda.\,\!</math>|4.5}}

We know from Eq.[[#eq4.4|(4.4)]] that this can be rewritten:

{{Equation|<math>P(\vec{a},\vec{b}) = -\int \rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda)d\lambda.\,\!</math>|4.6}}

Now for the clever part. Introducing another unit vector, <math>\vec{c}</math>, and noting that <math>[A(\vec{b},\lambda)]^{2}=1,\,\!</math>

{{Equation|<math>\begin{align}
P(\vec{a},\vec{b})-P(\vec{a},\vec{c}) &=
-\int \rho(\lambda)[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]d\lambda\\
&= -\int \rho(\lambda)[1-A(\vec{b},\lambda)A(\vec{c},\lambda)]A(\vec{a},\lambda)A(\vec{b},\lambda)d\lambda
\end{align}\,\!</math>|4.7}}

Recognizing some inequalities,

{{Equation|<math>\begin{align}
-1 \le A(\vec{a},\lambda)A(\vec{b},\lambda) \le 1, \\
\rho(\lambda)[1-A(\vec{b},\lambda)A(\vec{c},\lambda)] \ge 0,
\end{align}\,\!</math>|4.8}}

we get to a remarkable result,

{{Equation|<math>\begin{align}
|P(\vec{a},\vec{b})-P(\vec{a},\vec{c})| &\le \int \rho(\lambda)[1-A(\vec{b},\lambda)A(\vec{c},\lambda)]d\lambda \\
&\le 1 + P(\vec{b},\vec{c}). \end{align}\,\!</math>|4.9}}

The last form is known as the Bell inequality. This inequality is true for any local hidden variable theory.

What does this mean? Let us define <math>\vec{a}\,\!</math> and <math>\vec{b}\,\!</math> to be orthogonal and <math>\vec{c}\,\!</math> to make a <math>45^{\circ}\,\!</math> angle with both of them. Using quantum mechanics (Equation[[#eq4.3|(4.3)]]),

<math>\begin{align}
P(\vec{a},\vec{b}) = 0 \\
P(\vec{a},\vec{c}) = P(\vec{b},\vec{c}) = -\frac{1}{\sqrt{2}}.
\end{align} \,\!</math>

Inserting the values into the Bell inequality (Equation [[#eq4.9|(4.9)]]),

<math>\begin{align}
\left|-\frac{1}{\sqrt{2}} \right| &\le 1 + \left(-\frac{1}{\sqrt{2}}\right) \\
\frac{1}{\sqrt{2}} &\le \frac{\sqrt{2}-1}{\sqrt{2}} \\
1 &\le \sqrt{2} - 1
\end{align} \,\!</math>

Since <math>\sqrt{2}-1 \approx .41, \,\!</math> the inequality is violated!

This means that quantum mechanics is incompatible with ''any'' local hidden variable theory. The EPR paradox had stronger implications than the authors realized; if local realism is held, then quantum mechanics is incorrect. This has been repeatedly disproved experimentally. Thus no local hidden variable theory can resolve the "spooky-action-at-a-distance."

===Entangled Pure States===

Let us consider two quantum systems, one called A and the other B. Let us suppose the joint state of the entire system comprised of A and B is a [[Index#P|pure state]]. If the subsystems are independent and have never interacted, then the state of the composite system of the two particles can be written as
{{Equation|<math>
\left\vert \Phi\right\rangle = \left\vert \psi\right\rangle_A\otimes\left\vert \phi\right\rangle_B,
</math>|4.10}}
where <math>\left\vert \psi\right\rangle_A\,\!</math> is the state of particle A and <math>\left\vert \phi\right\rangle_B\,\!</math> is the state of subsystem B. This tensor product structure is sometimes stated as a postulate of quantum mechanics as in [[Bibliography#NielsenChuang:book|Nielsen and Chuang's book]]. In this case the two particles are not correlated in any way---they are said to be ''unentangled'' or ''separable''. When a pure state cannot be written in this form it is said to be ''entangled''.

For example, the most general form for a pure state of two qubits is given by Eq. [[Chapter 2 - Qubits and Collections of Qubits#eq2.30|(2.30)]].
<center>
<math>\left\vert{\psi}\right\rangle = \alpha_{00}\left\vert{00}\right\rangle + \alpha_{01}\left\vert{01}\right\rangle
+ \alpha_{10}\left\vert{10}\right\rangle + \alpha_{11}\left\vert{11}\right\rangle.</math>
</center>
Examples are given below of states that are entangled and thus cannot be written in the form of Eq. [[#eq4.10|(4.10)]].

For two particles (or systems) to become entangled, they must first interact with each other. This entanglement cannot increase (usually) by acting on an individual even both subsystems separately. Only joint measurements on both or interactions between the two can increase entanglement. Actions on an individual particle, without involving the other, are called ''local actions'' or ''local operations''.  For example, local unitary operations on individual particles can be written as
{{Equation|<math>
L = U_A\otimes U_B,
\,\!</math>|4.11}}
so that
{{Equation|<math>
L\left\vert \Phi\right\rangle = U_A\left\vert \psi\right\rangle_A\otimes U_B\left\vert \phi\right\rangle_B.
</math>|4.12}}<br />
Local unitary transformations will not change the entanglement of a system. Furthermore, local measurements or local measurements combined with unitary transformations cannot, on average, increase the entanglement between subsystems.

For later use, it is relevant to note that the density matrix for the composite system in Eq.[[#eq4.10|(4.10)]] is
{{Equation|<math>\begin{align}
\rho_{ss} &= \left\vert\Phi\right\rangle\left\langle\Phi\right\vert \\
&= (\left\vert \psi \right\rangle_A\otimes \left\vert \phi \right\rangle_B)(_A\left\langle \psi\right\vert\otimes_B\!\!\left\langle\phi\right\vert) \\
&= \left\vert \psi\right\rangle_A\left\langle \psi\right\vert\otimes \left\vert \phi\right\rangle_B\left\langle\phi\right\vert \\
&= \rho_A\otimes\rho_B,
\end{align}
\,\!</math>|4.13}}
where <math>\rho_A = \left\vert \psi\right\rangle_A\left\langle\psi\right\vert\,\!</math> and <math>\rho_B=
\left\vert \phi\right\rangle_B\left\langle \phi\right\vert\,\!</math> ---the density operator of a product state is
the product of density operators.

====Bell States====

The simplest examples of entangled states are the entangled states of
two two-state systems. There are four different versions of what is known as the
<nowiki>"maximally entangled state"</nowiki> of two qubits. The <nowiki>"maximally"</nowiki> will
be explained below. These four different versions are called Bell states and are
{{Equation|<math>\begin{align}
\left\vert \psi_+\right\rangle &= \frac{1}{\sqrt{2}}(\left\vert 01\right\rangle +\left\vert 10\right\rangle), \\
\left\vert \psi_-\right\rangle &= \frac{1}{\sqrt{2}}(\left\vert 01\right\rangle -\left\vert 10\right\rangle), \\
\left\vert \phi_+\right\rangle &= \frac{1}{\sqrt{2}}(\left\vert 00\right\rangle +\left\vert 11\right\rangle), \\
\left\vert \phi_-\right\rangle &= \frac{1}{\sqrt{2}}(\left\vert 00\right\rangle -\left\vert 11\right\rangle).
\end{align}
</math>|4.14}}
This is an orthonormal set of states that are all able to be obtained from each other by acting on one particle alone or both individual particles with unitary transformations (i.e. acting with local unitary transformations). For example, consider the local unitary transformation <math>\mathbb{I}\otimes\sigma_3\,\!</math> acting on <math>\left\vert \phi_-\right\rangle\,\!</math>. The result is <math>\left\vert \phi_+\right\rangle\,\!</math>. Similarly, <math>\sigma_1\otimes\mathbb{I}\,\!</math> acting on <math>\left\vert \psi_+\right\rangle\,\!</math> yields <math>\left\vert \phi_+\right\rangle\,\!</math>, and so on.

These states certainly cannot be written in the form
<center><math>
\left\vert \Phi\right\rangle = \left\vert \psi\right\rangle_A\otimes\left\vert \phi\right\rangle_B.
\,\!</math></center>
What if they could? Let
<math>\left\vert \psi\right\rangle_A=\alpha_0\left\vert 0\right\rangle+\alpha_1\left\vert 1\right\rangle\,\!</math> and
<math>\left\vert \phi\right\rangle_B=\beta_0\left\vert 0\right\rangle+\beta_1\left\vert 1\right\rangle\,\!</math>. Notice that the general
form is
{{Equation|<math>\begin{align}
\left\vert \psi\right\rangle_A\otimes\left\vert \phi\right\rangle_B &=
(\alpha_0\left\vert 0\right\rangle+\alpha_1\left\vert 1\right\rangle)\otimes(\beta_0\left\vert 0\right\rangle+\beta_1\left\vert 1\right\rangle), \\
&= \alpha_0\beta_0\left\vert 00\right\rangle +\alpha_1\beta_0\left\vert 10\right\rangle
+ \alpha_0\beta_1\left\vert 01\right\rangle +\alpha_1\beta_1\left\vert 11\right\rangle,
\end{align}
</math>|4.15}}
so the coefficient of <math>\left\vert 00\right\rangle\,\!</math> times the coefficient of <math>\left\vert 11\right\rangle\,\!</math> minus the coefficient of <math>\left\vert 01\right\rangle\,\!</math> times the coefficient of <math>\left\vert 10\right\rangle\,\!</math> is zero. This is not true for any of the Bell states---thus, they cannot be written as a tensor product of two 1-particle states. So, for any 2-particle state,
{{Equation|<math>
\left\vert \Psi\right\rangle = \alpha_{00} \left\vert 00\right\rangle + \alpha_{01} \left\vert 01\right\rangle
+ \alpha_{10} \left\vert 10\right\rangle + \alpha_{11} \left\vert 11\right\rangle,
</math>|4.16}}<br />
the state is separable or unentangled if (and only if!) <math>\alpha_{00}\alpha_{11} -
\alpha_{01}\alpha_{10} = 0\,\!</math>. Otherwise, it is entangled.

===Entangled Mixed States===

The state in Eq.[[#eq4.1|(4.1)]] is not entangled, so it is called ''separable''. More precisely it is referred to as a simply separable state.  In general, a state is separable if its density matrix can be written in the form 
{{Equation|<math>
\rho_s = \sum_i p_i \rho_A^{(i)} \otimes \rho_B^{(i)},
</math>|4.17}}<br />
where <math>\rho_A^{(i)}\,\!</math> <math>(\rho_B^{(i)})\,\!</math> is a valid density matrix for
subsystem <math>A\,\!</math> <math>(B)\,\!</math>, and <math>\sum_ip_i =1, \; p_i\geq 0\,\!</math>.
An ''entangled state''  is one that cannot be written in the form of Eq.[[#eq4.17|(4.17)]].

For a pure state, the situation is simpler. A pure state is entangled if and only if it cannot be written in the form of
Eq.[[#eq4.10|(4.10)]]. In other words a pure state is entangled if it cannot be written as the product of two states of the individual
subsystems.

====Reduced Density Operators and the Partial Trace====


The Bell states are maximally entangled states.  To understand this, one may consider the fact that these states are pure states but information about the individual particles in the system is lacking. In this section, a
more precise meaning of this statement is given.

Let us first consider useful tool, the ''partial trace''.  The partial trace is the
trace over one of the subsystems (particle states) of a composite system. Let us suppose that the density matrix for a composite system
is given by
{{Equation|<math>
\rho_{ss} = \rho_A\otimes\rho_B.
\,\!</math>|4.18}}<br />
The partial trace  is the trace over one of the subsystems. For example, the trace over subsystem <math>B\,\!</math> is
{{Equation|<math>
\mbox{Tr}_B(\rho_{ss}) = \rho_A\mbox{Tr}(\rho_B) = \rho_A,
\,\!</math>|4.19}}<br />
since <math>\mbox{Tr}(A\otimes B) =\mbox{Tr}(A)\mbox{Tr}(B)\,\!</math> and the trace of a density matrix is one. The matrix <math>\rho_A\,\!</math> is called the ''reduced density operator'',  or reduced density matrix.

However, this is a special case. The density matrix for a composite system of two (or more) subsystems cannot be written in this form except in very special circumstances---when the two subsystems have never interacted and there are no correlations between them.

For the cases where the two subsystems are entangled, there are at
least two ways to calculate the partial trace. One is to write
the matrix form of the state in terms of a sum of the tensor products of Pauli
matrices. (See [[Appendix E - Density Operator: Extensions]], Sec. [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|Two-State Example: Bloch Sphere]].)
The other is to realize that the trace can be calculated by
summing the projections onto the diagonal elements of the subsystem
over which you are tracing. For example, for
a general <math>2\times 2\,\!</math> density matrix <math>\rho_2\,\!</math>, the trace is
{{Equation|<math>
\mbox{Tr}(\rho_2) = \mbox{Tr}(\left\vert 0\right\rangle\left\langle 0\right\vert\rho_2 +\left\vert 1\right\rangle\left\langle 1\right\vert\rho_2)
= \left\langle 0\right\vert\rho_2\left\vert 0\right\rangle + \left\langle 1\right\vert\rho_2\left\vert 1\right\rangle.
</math>|4.20}}<br />

For the general case, let us consider a density matrix for a bipartite
system, <math>\rho\,\!</math>. Let the subsystem <math>A\,\!</math> have Greek letters as indices and
let the subsystem <math>B\,\!</math> have Latin indices. Then
{{Equation|<math>\begin{align}
\rho &= \sum_{\alpha,\beta,i,j} \rho_{\alpha\beta,ij} \left\vert \alpha
i\right\rangle\left\langle\beta j\right\vert \\
&= \sum_{\alpha,\beta,i,j} \rho_{\alpha\beta,ij} \left\vert \alpha\right\rangle_A \otimes
\left\vert i\right\rangle_{B \; A}\!\left\langle\beta \right\vert \otimes{}_B\!\left\langle j\right\vert \\
&= \sum_{\alpha,\beta,i,j} \rho_{\alpha\beta,ij}
\left\vert\alpha\right\rangle_{A \; A}\!\left\langle\beta\right|\otimes\left| i\right\rangle_{B \; B}\!\left\langle j\right|.
\end{align}
</math>|4.21}}
To calculate the reduced density matrix of subsystem <math>A\,\!</math> by tracing over subsystem <math>B\,\!</math>, the trace
over <math>B\,\!</math> is taken by computing 
{{Equation|<math>\begin{align}
\rho_A &= \mbox{Tr}_B(\rho) \\
&= \sum_{k} {}_B\!\left\langle k\right\vert\rho\left\vert k\right\rangle_B \\
&= \sum_{k} \sum_{\alpha,\beta,i,j} {}_B\!\left\langle k\right\vert \left(\rho_{\alpha\beta,ij}
\left\vert\alpha\right\rangle_{A\; A}\!\left\langle\beta\right\vert\otimes\left\vert i\right\rangle_{B\; B}\!\left\langle j\right\vert\right) \left\vert k\right\rangle_B \\
&= \sum_{k} \sum_{\alpha,\beta,i,j}\rho_{\alpha\beta,ij}
\left\vert\alpha\right\rangle_{A\; A}\!\left\langle\beta\right\vert\delta_{ki}\delta_{jk} \\
&= \sum_{\alpha,\beta,k} \rho_{\alpha\beta,kk}
\left\vert\alpha\right\rangle_{A\; A}\!\left\langle\beta\right\vert \\
\end{align}
</math>|4.22}}

For the partial trace of a <math>4\times 4\,\!</math> density matrix <math>\rho\,\!</math> over the
subsystem <math>B\,\!</math>,
{{Equation|<math>
\mbox{Tr}_B(\rho) = {}_B\left\langle 0\right\vert \rho \left\vert 0\right\rangle_B
+ {}_B\left\langle 1\right\vert\rho\left\vert 1\right\rangle_B,
</math>|4.23}}<br />
which leaves the part of the matrix corresponding to <math>A\,\!</math> alone
and projects the <math>B\,\!</math> part onto the two diagonal elements and then adds
those. Now let us calculate the partial trace of a Bell state, for example
<math>\left\vert \psi_+\right\rangle\,\!</math>. Assuming the first state is the <math>A\,\!</math> state and the second <math>B\,\!</math>, we see

{{Equation|<math>\begin{align}
\mbox{Tr}_B(\left\vert\psi_+ \right\rangle\left\langle \psi_+\right\vert) &=
{}_B\left\langle 0\right\vert (\left\vert\psi_+ \right\rangle\left\langle\psi_+\right\vert)\left\vert 0\right\rangle_B
+ {}_B\left\langle 1\right\vert (\left\vert\psi_+\right\rangle\left\langle \psi_+\right\vert)\left\vert 1\right\rangle_B, \\
&= {}_B\left\langle 0\right\vert\left(\frac{1}{2}(\left\vert 01\right\rangle +\left\vert 10\right\rangle)
(\left\langle 01\right\vert +\left\langle 10\right\vert)\right)\left\vert 0\right\rangle_B \\
& \;\; + {}_B\left\langle 1\right\vert\left(\frac{1}{2}\left(\left\vert 01\right\rangle +\left\vert 10\right\rangle\right)
(\left\langle 01\right\vert +\left\langle 10\right\vert)\right)\left\vert 1\right\rangle_B \\
&=\frac{1}{2}(\left\vert 1\right\rangle_A\left\langle 1\right\vert)
+\frac{1}{2}(\left\vert 0\right\rangle_A\left\langle 0\right\vert),
\end{align}
</math>|4.24}}
which can be rewritten simply as
{{Equation|<math>
\mbox{Tr}_B(\left\vert \psi_+\right\rangle \left\langle \psi_+\right\vert ) = \rho_A
= \left(\begin{array}{cc}
1/2 & 0 \\
0 & 1/2 \end{array}\right).
</math>|4.25}}<br />
This is quite an interesting and significant find. The density matrix
for the whole system of two qubits is in a pure state indicating
maximal knowledge. However, the reduced density matrix, representing
our knowledge of one of the individual particles, is completely (or
maximally) mixed, indicating minimal knowledge. This means that the two particles or subsystems taken together are in a definite, pure state; yet when taken separately, they contain as little information as possible. This fact indicates entanglement.

It is important to note that for a pure state, the trace over
subsystem <math>A\,\!</math> produces the same result as the trace over subsystem
<math>B\,\!</math>. In other words, for a pure state <math>\rho_p\,\!</math>,
{{Equation|<math>
\mbox{Tr}_A(\rho_p) = \mbox{Tr}_B(\rho_p).
\,\!</math>|4.26}}<br />

====How Entangled Is It?====

The previous discussion showed that there is a definite notion of maximal entanglement for pure states. From the determinant condition, there is a method for identifying unentangled pure states. A question may arise: how entangled is it if it is not separable nor maximally entangled? There are now many ways of defining measures of entanglement which will be explored in a later section and an appendix. Here, a common way of measuring the entanglement for pure states is given which is based on the partial trace of a pure state.

Let us consider the extreme cases. First, we found that if the states are Bell states, then the partial trace of the bipartite system of two qubits will yield a density operator for the subsystem that is maximally entangled. Due to the fact that the trace of the density operator is always one, it should be clear that the partial trace of a separable pure state gives a pure state density operator. Notice that the purity of the partial density operator provides a candidate for a measure of entanglement. Before we discuss this further, let us note the following important result, called the Schmidt decomposition.

'''Schmidt Decomposition'''

For any pure state density operator of a bipartite system, say with constituents A and B, it can be taken as
{{Equation|<math>
\left\vert \Psi\right\rangle_{AB} = \sum_{ij} a_{ij}^\prime\left\vert\psi_i^\prime\right\rangle_A\otimes\left\vert\phi_j^\prime\right\rangle_B.
\,\!</math>|4.27}}<br />
This can also be put in the form
{{Equation|<math>
\left\vert\Psi\right\rangle_{AB} = \sum_i a_i \left\vert\psi_i\right\rangle_A\otimes\left\vert\phi_i\right\rangle_B.
\,\!</math>|4.28}}<br />

In this case it is clear that the reduced density operator is the same for each subsystem and thus the purity of the reduced density operator can be used as a measure of entanglement.

===Extensions and Open Problems===

[[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Introduction|Continue to '''Chapter 5 - Quantum Information: Basic Principles and Simple Examples]]

Appendix F - Classical Error Correcting Codes

2011-11-07T16:46:41Z

Stempel2: /* The Hamming Bound */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall that we are working with binary codes. Thus the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets, with each containing only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset that is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math>, and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, then this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors, with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is, <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code that can correct all errors up to those of weight <math> t \,\!</math>, and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant that goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math>, is the set of all vectors that have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors, a vector can be orthogonal to itself. Note that this is different from ordinary vectors in 3-d space.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to the rate of an error correcting code. This does not indicate whether or not codes that satisfy these bounds exist, but it does tell us that no codes exist that do not satisfy these bounds. Encoding, decoding, error detection and correction are all difficult problems to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a look-up table. This would be much more time-consuming than using the parity check matrix; matrix multiplication is quite efficient relative to the look-up table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues---as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary due to the delicacy of quantum information. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-07T16:45:48Z

Stempel2: /* Generator Matrix */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall that we are working with binary codes. Thus the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets, with each containing only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset that is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math>, and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, then this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors, with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code that can correct all errors up to those of weight <math> t \,\!</math>, and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant that goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math>, is the set of all vectors that have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors, a vector can be orthogonal to itself. Note that this is different from ordinary vectors in 3-d space.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to the rate of an error correcting code. This does not indicate whether or not codes that satisfy these bounds exist, but it does tell us that no codes exist that do not satisfy these bounds. Encoding, decoding, error detection and correction are all difficult problems to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a look-up table. This would be much more time-consuming than using the parity check matrix; matrix multiplication is quite efficient relative to the look-up table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues---as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary due to the delicacy of quantum information. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-07T16:44:50Z

Stempel2: /* Final Comments */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets, with each containing only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset that is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math>, and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, then this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors, with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code that can correct all errors up to those of weight <math> t \,\!</math>, and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant that goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math>, is the set of all vectors that have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors, a vector can be orthogonal to itself. Note that this is different from ordinary vectors in 3-d space.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to the rate of an error correcting code. This does not indicate whether or not codes that satisfy these bounds exist, but it does tell us that no codes exist that do not satisfy these bounds. Encoding, decoding, error detection and correction are all difficult problems to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a look-up table. This would be much more time-consuming than using the parity check matrix; matrix multiplication is quite efficient relative to the look-up table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues---as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary due to the delicacy of quantum information. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-07T16:41:24Z

Stempel2: /* Definition 11: Dual Code */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets, with each containing only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset that is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math>, and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, then this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors, with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code that can correct all errors up to those of weight <math> t \,\!</math>, and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant that goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math>, is the set of all vectors that have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors, a vector can be orthogonal to itself. Note that this is different from ordinary vectors in 3-d space.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-07T16:39:19Z

Stempel2: /* The Hamming Bound */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets, with each containing only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset that is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math>, and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, then this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors, with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code that can correct all errors up to those of weight <math> t \,\!</math>, and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant that goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-07T16:36:09Z

Stempel2: /* The Disjointness Condition and Correcting Errors */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets, with each containing only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset that is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math>, and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, then this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors, with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-04T15:37:45Z

Stempel2: /* The Disjointness Condition and Correcting Errors */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which containing only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset that is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math>, and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, then this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T18:36:07Z

Stempel2: /* Example 3 */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
(See for example [[Bibliography#LoeppWootters|Loepp and Wootters]].) From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors that are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix is <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. Only the codewords are annihilated by the parity check matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T18:30:44Z

Stempel2: /* Errors */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions that must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected; here, the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original (logical or encoded) state <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring:
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords; i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T18:26:44Z

Stempel2: /* Parity Check Matrix */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix, <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>: <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. It follows that
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results, <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math>, must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T18:18:05Z

Stempel2: /* Generator Matrix */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix,''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis that will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math> described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched or even added to produce a new vector that replaces a column, then the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent, which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T17:11:14Z

Stempel2: /* Definition 7 */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code that is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis which will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math>, described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched, or even added to produce a new vector which replaces a column, the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T17:10:49Z

Stempel2: /* Example 2 */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be
{{Equation|<math>
0_L = 000 \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state; that is, one that is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words, and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code which is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis which will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math>, described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched, or even added to produce a new vector which replaces a column, the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T16:40:24Z

Stempel2: /* Example 1 */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2, which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be the following
{{Equation|<math>
0_L = 000, \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state, that is, one which is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code which is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis which will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math>, described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched, or even added to produce a new vector which replaces a column, the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T16:38:44Z

Stempel2: /* Definition 4 */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> to denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors, but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2 which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be the following
{{Equation|<math>
0_L = 000, \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state, that is, one which is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code which is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis which will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math>, described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched, or even added to produce a new vector which replaces a column, the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T15:19:23Z

Stempel2: /* Definition 3 */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>. Then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors, but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2 which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be the following
{{Equation|<math>
0_L = 000, \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state, that is, one which is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code which is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis which will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math>, described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched, or even added to produce a new vector which replaces a column, the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T15:16:17Z

Stempel2: /* Definition 1 */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree, or have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the other vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>, then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors, but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2 which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be the following
{{Equation|<math>
0_L = 000, \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state, that is, one which is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code which is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis which will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math>, described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched, or even added to produce a new vector which replaces a column, the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix F - Classical Error Correcting Codes

2011-11-02T15:05:05Z

Stempel2: /* Binary Operations */

===Introduction===

Classical error correcting codes are in use in a wide variety of digital electronics and other classical information systems. It is a good idea to learn some of the basic definitions, ideas, methods, and simple examples of classical error correcting codes in order to understand the (slightly) more complicated quantum error correcting codes. There are many good introductions to classical error correction. Here we follow a few sources which also discuss quantum error correcting codes: the book by [[Bibliography#LoeppWootters|Loepp and Wootters]], an article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane, [[Bibliography#GottDiss|Gottesman's Thesis]], and [[Bibliography#Gaitan:book|Gaitan's Book]] on quantum error correction, which also discusses classical error correction.

===Binary Operations===

The set <math> \{0,1\} \,\!</math> is a group under addition. (See [[Appendix D - Group Theory#Example 3|Section D.2.8]] of [[Appendix D - Group Theory|Appendix D]].) The way this is achieved is by deciding that we will only use these two numbers in our language and using addition modulo 2, meaning <math> 0+0=0, 1+0 = 0+1 = 1, \,\!</math> and <math>1+1 =0\,\!</math>. If we also include the operation of multiplication, the set (with the two operations) becomes a field (a Galois Field), which is denoted GF<math>(2)\,\!</math>. Since one often works with strings of bits, it is very useful to consider the string of bits to be a vector and to use vector addition (which is component-wise addition) and vector multiplication (which is the inner product). For example, the addition of the vector <math>(0,0,1)\,\!</math> and <math>(0,1,1)\,\!</math> is <math>(0,0,1) + (0,1,1) = (0,1,0)\,\!</math>. The inner product between these two vectors is <math>(0,0,1) \cdot (0,1,1) = 0\cdot 0 + 0\cdot 1 + 1\cdot 1 = 0 +0 +1=1\,\!</math>.

===Definitions and Basics===

====Definition 1====
The inner product is also called a '''checksum''' or '''parity check''' since it shows whether or not the first and second vectors agree have an even number of 1's at the positions specified by the ones in the other vector. We may say that the first vector satisfies the parity check of the first vector, or vice versa.

====Definition 2====
The '''weight''' or '''Hamming weight''' is the number of non-zero components of a vector or string. The weight of a vector <math>v\,\!</math> is denoted wt(<math>v\,\!</math>).

====Definition 3====
The '''Hamming distance''' is the number of places where two vectors differ. Let the two vectors be <math>v\,\!</math> and <math>w\,\!</math>, then the Hamming distance is also equal to wt(<math>v+w\,\!</math>). The Hamming distance between <math>v\,\!</math> and <math>w\,\!</math> will be denoted <math>d_H(v,w)\,\!</math>.

====Definition 4====
We use <math>\{0,1\}^n\,\!</math> denote the set of all binary vectors of length <math>n\,\!</math>. A '''code''' <math>C\,\!</math> of length <math>n\,\!</math> is any subset of that set. The set of all elements of <math>C\,\!</math> is called the set of '''codewords'''. We also say there are <math>2^n\,\!</math> <math>n\,\!</math>-bit words in the space.

Suppose <math>n\,\!</math> bits are used to encode <math>k\,\!</math> logical bits. We use the notation <math>[n,k] \,\!</math> do denote such a code.

====Definition 5====
The '''minimum distance''' of a code is the smallest Hamming distance between any two non-equal vectors in a code. This can be written
{{Equation|<math>
d_{Hmin}(C) = \underset{v,w\in C,v\neq w}{\mbox{min}}d_H(v,w).
\,\!</math>|F.1}}
For shorthand, we also use <math> d(C)\,\!</math> or <math> d\,\!</math> if <math> C\,\!</math> is understood.

When that code has a distance <math>d\,\!</math>, the notation <math>[n,k,d] \,\!</math> is used.

====Example 1====
It is interesting to note that if we encode redundantly using <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math> as our logical zero and logical one respectively, then we could detect single bit errors, but not correct them. For example, if we receive <math> 01\,\!</math>, we know this cannot be one of our encoded states. So an error must have occurred. However, we don't know whether the sender sent <math> 0_L=00 \,\!</math> or <math>1_L=11\,\!</math>. We do know that an error has occurred though, as long as we know only one error has occurred. Such an encoding can be used as an '''error detecting code'''. In this case there are two code words, <math> 0_L=00 \,\!</math> and <math>1_L=11\,\!</math>, but four words in the space. The minimum distance is 2 which is the distance between the two code words.

====Example 2====
The three-bit redundant encoding was already given in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]]. One takes logical zero and logical one states to be the following
{{Equation|<math>
0_L = 000, \;\;\; \mbox{ and } \;\;\; 1_L = 111,
\,\!</math>|F.2}}
where the subscript <math>L \,\!</math> is used to denote a "logical" state, that is, one which is encoded. Recall that this code is able to detect and correct one error. In this case there are two code words out of eight possible words and the minimal distance is 3.

====Definition 6====
The '''rate''' of a code is given by the ration of the number of logical bits to the number of bits, <math>k/n\,\!</math>.

====Definition 7====
A '''linear code''' <math>C_l\,\!</math> is a code which is closed under addition.

===Linear Codes===

Linear codes are particularly useful because they are able to efficiently identify errors and the associated correct codewords. This ability is due to the added structure these codes have. These will be discussed in the following sections.

====Generator Matrix====

For linear codes, any linear combination of codewords is a codeword. One key feature of a linear code is that it can be specified by a <nowiki>''generator matrix''</nowiki> <math>G\,\!</math><ref>Recall we are working with binary codes. So the entries of the matrix will also be binary numbers, i.e., 0's and 1's.</ref>. For an <math> [n,k]\,\!</math> code, the '''generator matrix''' is an <math> n\times k\,\!</math> matrix with columns that form a basis for the <math>k\,\!</math>-dimensional coding sub-space of the <math>n\,\!</math>-dimensional binary vector space. In other words, the vectors comprising the rows form a basis which will span the code space. (Note that one may also use the transpose of this matrix as the definition for <math>G\,\!</math>.) Any code word <math>w\,\!</math>, described by a vector <math>v\,\!</math> can be written in terms of the generator matrix as <math>w = Gv\,\!</math>. Note that <math>G\,\!</math> is independent of the input and output vectors. In addition, <math>G\,\!</math> is not unique. If columns are switched, or even added to produce a new vector which replaces a column, the generator matrix is still valid for the code. This is due to the requirement that the columns be linearly independent which is still satisfied if these operations are performed.

====Parity Check Matrix====
Once <math>G\,\!</math> is obtained, one can calculate another useful matrix <math>P\,\!</math>. <math>P\,\!</math> is an <math>(n\times k)\times n\,\!</math> matrix which has the property that
{{Equation|<math>
PG = 0.
\,\!</math>|F.3}}
The matrix <math>P\,\!</math> is called the '''parity check matrix''' or '''dual matrix'''. The rank of <math>P\,\!</math> is at most <math>n- k\,\!</math> and has the property that it annihilates any code word. To see this, recall any code word is written as <math>Gv\,\!</math>, so <math>PGv =0\,\!</math> since <math>PG =0\,\!</math>. Also, due to the rank of <math>P\,\!</math>, it can be shown that <math>Pw =0\,\!</math> only if <math>w\,\!</math> is a code word. That is to say, <math>Pw=0\,\!</math> if and only if <math>w\,\!</math> is a code word. This means that <math>P\,\!</math> can be used to test whether or not a word is in the code.

Suppose an error occurs on a code word <math>w\,\!</math> to produce <math>w^\prime = w + e\,\!</math>. Now consider
{{Equation|<math>
Pw^\prime = P(w+e) = Pe,
\,\!</math>|F.4}}
since <math>Pw=0\,\!</math>. This result, <math>Pe\,\!</math> is called the '''error syndrome''' and the measurement to identify <math>Pe\,\!</math> is called the '''syndrome measurement'''. Therefore, the result depends only on the error and not on the original code word. If the error can be determined from this result, then it can be corrected independent of the code word. However, in order to have <math>Pe\,\!</math> be unique, two different results <math>Pe_1\,\!</math> and <math>Pe_2\,\!</math> must not be equal. This is possible if a distance <math>d\,\!</math> code is constructed such that the parity check matrix has <math>d-1=2t\,\!</math> linearly independent columns. This enables the errors to be identified and thus corrected.

===Errors===

For any classical error correcting code, there are general conditions which must be satisfied in order for the code to be able to detect and correct errors. The two examples above show how the error can be detected, but here the objective is to give some general conditions.

Note that any state containing an error may be written as the sum of the original state (logical or encoded state) <math>w \,\!</math> and another vector <math>e \,\!</math>. The error vector <math>e \,\!</math> has ones are in the places where errors are present and zeroes everywhere else. To ensure that the error may be corrected, the following condition must be satisfied for two states with errors occurring
{{Equation|<math>
w_1 + e_1 \neq w_2 + e_2.
\,\!</math>|F.5}}
This condition is called the '''disjointness condition'''. This condition means that an error on one state cannot be confused with an error on another state. If it could, then the state including the error could not be uniquely identified with an encoded state and the state could not be corrected to its original state before the error occurred. More specifically, for a code to correct <math>t\,\!</math> single-bit errors, it must have distance at least <math>2t + 1 \,\!</math> between any two codewords, i.e., it must be a distance <math>2t + 1 \,\!</math> code, i.e., it must be true that <math>d(C) \geq 2t + 1 \,\!</math>. An <math>[n,k]\,\!</math> code with minimal distance <math>d \,\!</math> is denoted <math>[n,k,d]\,\!</math>.

====Example 3====
An important example of an error correcting code is called the <math>[7,4,3]</math> Hamming code. This code, as the notation indicates, encodes <math>k=4</math> bits of information into <math>n=7</math> bits. It also does it in such a way that one error can be detected and corrected since it has a distance of <math>3</math>. The generator matrix for this code can be taken to be (see for example [[Bibliography#LoeppWootters|Loepp and Wootters]])
{{Equation|<math>
G^T = \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 1 & 1
\end{array}\right).
\,\!</math>|F.6}}
From this the parity check matrix, <math>P\,\!</math> can be calculated by finding a set of <math>n-k\,\!</math> mutually orthogonal vectors which are also orthogonal to the code space defined by the generator matrix. Alternatively, one could find the generator matrix from the parity check matrix. A method for doing this can be found in Steane's article in [[Bibliography#LoPopescuSpiller|Lo, Popescu, and Spiller]]. One first puts <math>G^T\,\!</math> in the form <math>(I_k,A),\,\!</math> where <math>I_k\,\!</math> is the <math>k\times k\,\!</math> identity matrix. Then the parity check matrix <math>P = (A^T,I_{n-k}).\,\!</math> In either case, one can arrive at the following parity check matrix for this code:
{{Equation|<math>
P = \left(\begin{array}{ccccccc}
1 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 1
\end{array}\right).
\,\!</math>|F.7}}
It is useful to note that the code can also be defined by the parity check matrix. The codewords are annihilated by the parity check matrix and only the codewords are annihilated by that matrix.

===The Disjointness Condition and Correcting Errors===

The motivation for the disjointness condition, [[#eqF.5|Eq.(F.5)]], is to associate each vector in the space with a particular code word. That is, assuming that only certain errors occur, each error vector should be associated to a particular vector in the code space when the error is added to the original code word. This partitions the set into disjoint subsets each of which contains only one code vector. A message is decoded correctly if the vector (the one containing the error) is in the subset which is associated with the original vector (the one with no error). For example, if one vector is sent, say <math> v_1 \,\!</math> and an error occurs during transmission to produce <math> v_2 = v_1 +e\,\!</math>, this vector must be in the subset containing <math> v_1 \,\!</math>.

A way to decode is to record an array of possible code words, possible errors, and the combinations of those errors and code words. The array can be set up as a top row of the code word vectors and a leftmost column of errors with the element of the first row and the first column being the zero vector and all subsequent entries in the column being errors. Then the element at the top of a column (say the jth column) is added to the error in the corresponding row (say the kth row) to get the j,k entry of the array. With this array one can associate a column with a subset that is disjoint with the other sets. Then identifying the erred code word in a column associates it with a code word and thus corrects the error.

===The Hamming Bound===

The Hamming bound is a bound that restricts the rate of the code. Due to the disjointness condition, a certain number of bits are required to ensure our ability to detect and correct errors. Suppose there is a set of <math> n\,\!</math> bit vectors for encoding <math> k\,\!</math> bits of information. There is a set of error vectors of weight <math> t \,\!</math> that has <math> C(n,t)\,\!</math> elements<ref>That is <math> n \,\!</math> choose <math> t \,\!</math> vectors. The notation is <math> C(n,t) = {n\choose t} = \frac{n!}{(n-t)!t!}.\,\!</math></ref>. So the number of error vectors, including errors of weight up to <math> t \,\!</math>, is
<math> \sum_{i=0}^t C(n,i). \,\!</math> (Note that no error is also part of the set of error vectors. The objective is to be able to design a code which can correct all errors up to those of weight <math> t \,\!</math> and this includes no error at all.) Since there are <math> 2^n\,\!</math> vectors in the whole space of <math> n\,\!</math> bits, and assuming <math> m\,\!</math> vectors are used for the encoding, the Hamming bound is
{{Equation|<math>
m\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.8}}
For linear codes, <math> m=2^k,\,\!</math> so
{{Equation|<math>
2^k\sum_{i=0}^t C(n,i) \leq 2^n.
\,\!</math>|F.9}}
Taking the logarithm,
{{Equation|<math>
k \leq n - \log_2\left(\sum_{i=0}^t C(n,i)\right).
\,\!</math>|F.10}}
For large <math> n, k \,\!</math> and <math> t \,\!</math>, we can use [[#LoPopescueSpiller|Stirling's formula]] to show that
{{Equation|<math>
\frac{k}{n} \leq 1 - H\left(\frac{t}{n}\right),
\,\!</math>|F.11}}
where <math> H(x) = -x\log x -(1-x)\log (1-x) \,\!</math> and we have neglected an overall multiplicative constant which goes to 1 as <math> n\rightarrow \infty. \,\!</math> (Again, see the article in [[Bibliography#LoPopescueSpiller|Lo, Popescu, and Spiller]] by Steane.)

===More Definitions===

====Definition 11: Dual Code====

Let <math>\mathcal{C}\,\!</math> be a code and let <math>v\,\!</math> be a vector in the code space. The '''dual code''', denoted <math>\mathcal{C}^\perp\,\!</math> is the set of all vectors which have zero inner product with all <math>v\in \mathcal{C}\,\!</math>. In other words, it is the set of all vectors <math>u\,\!</math> such that <math>u\cdot v = 0,\,\!</math> for all <math>v\in \mathcal{C}\,\!</math>.

For binary vectors a vector can be orthogonal to itself. This is different from ordinary vectors (in 3-d space) so this is one concept which is quite different in that regard.

The dual code is a useful entity in classical error correction and will be used in the construction of the quantum error correcting codes known as [[Chapter 7 - Quantum Error Correcting Codes#CSS codes|CSS codes]].

===Final Comments===

As can be seen from the Hamming bound, there is a limit to rate of an error correcting code. This does not indicate whether or not codes satisfying these bounds exists, but it does tell us that no codes exist which do not satisfy these bounds. Encoding decoding and error detection and correction are all problems which can be difficult to solve in general. One of the advantages of the linear codes is that they provide a systematic method for identifying errors on a code through the use of the parity check operation. More generally, checking to see whether or not a bit string (vector) is in the code space would require a lookup table. This would be much more time-consuming that using the parity check matrix where matrix multiplication is quite efficient relative to the lookup table.

Many of these ideas and definitions will be utilized in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]] on quantum error correction. Some linear codes, including the Hamming code above, will have quantum analogues as do many quantum error correcting codes. In quantum computers, as will be discussed, error correction is necessary since the quantum information is delicate. Such discussions will be taken up in [[Chapter 7 - Quantum Error Correcting Codes|Chapter 7]].

==Footnotes==
<references />

Appendix E - Density Operator: Extensions

2011-11-02T01:59:55Z

Stempel2: /* An N-dimensional Generalization of the Polarization Vector */

===Introduction===

The Bloch sphere picture for two state systems given in Section [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]] is quite useful. This Appendix presents a generalization to higher dimensions.

===An N-dimensional Generalization of the Polarization Vector===

The following are somewhat standard conventions and those contained in
Ref.~\cite{Byrd/Khaneja:03}. A
density operator on an <math>N\,\!</math>-dimensional Hilbert space <math>{\mathcal{H}}_N\,\!</math>
will be represented using a set of traceless Hermitian
matrices <math>\{\lambda_i\}\,\!</math>, <math>i=1,2,...,N^2-1\,\!</math> with the normalization
condition <math>\mbox{Tr}(\lambda_i\lambda_j)=2\delta_{ij}\,\!</math>. The commutation
and anticommutation relations for this set of matrices are given by
{{Equation|<math>
[\lambda_i,\lambda_j] = 2ic_{ijk}\lambda_k, \;\;\;
\{\lambda_i,\lambda_j\} = \frac{4}{N}I \delta_{ij} + 2d_{ijk}\lambda_k,
\,\!</math>|E.1}}
where the sum over repeated indices is to be understood unless
otherwise stated. (In some cases the sum is displayed explicitly for
emphasis.)
These relations can be summarized using the trace, antisymmetric, and
symmetric combinations of the following equation:
{{Equation|<math>
\lambda_i\lambda_j = \frac{2}{N}I \delta_{ij}
+ d_{ijk}\lambda_k + ic_{ijk}\lambda_k.
\,\!</math>|E.2}}
The density operator can now be written as
{{Equation|<math>
\rho = \frac{1}{N}\left(I + b \vec{n} \cdot \vec{\lambda}\right)
\,\!</math>|E.3}}
where <math>b = \sqrt{(N(N-1)/2)}\,\!</math>. The ''dot'' product is a sum
over repeated indices,
{{Equation|<math>
\vec{a}\cdot \vec{b} = a_ib_i = \sum_{i=1}^{N^2-1}a_ib_i.
\,\!</math>|E.4}}
Any complete set of <math>N^2-1\,\!</math> mutually trace-orthogonal, Hermitian matrices can serve as a basis and can be chosen to satisfy the conditions given here.

Using the condition that pure states satisfy <math>\rho^2 = \rho\,\!</math>, we find that for pure states,
{{Equation|<math>
\vec{n}\cdot\vec{n} = 1, \;\;\;\mbox{and} \;\;\; \vec{n}\star\vec{n} = \vec{n},
\,\!</math>|E.5}}
where the <nowiki>"star"</nowiki> product is defined by
{{Equation|<math>
(\vec{a}\star\vec{b})_k = \frac{1}{N-2}\sqrt{\frac{N(N-1)}{2}}\;d_{ijk}a_ib_j.
\,\!</math>|E.6}}
For later use, a <nowiki>"cross"</nowiki> product between two coherence vectors can
also be defined by
{{Equation|<math>
(\vec{a}\times\vec{b})_k = c_{ijk}a_ib_j.
\,\!</math>|E.7}}

====The Density Matrix for Two Qubits====

Chapter 3 - Physics of Quantum Information

2011-11-02T00:49:14Z

Stempel2: /* Two-State Example: Bloch Sphere */

===Introduction===

It was a great realization that information is physical and that a
(classical) Turing machine is not the end of the story of
computation. The physical system in which the information is stored
and manipulated is important and qubits are quite different from
bits.

In this chapter, some background in quantum mechanics is provided.
Not all of this chapter will be directly relevant to our discussion,
but it is included to progress our understanding
of how quantum mechanics from a textbook is related to quantum
computing. The connection is clear, but the story seems
incomplete from a physicists perspective. For the subject of error
prevention methods, some of this chapter will be vital---in
particular, the section(s) concerning the density matrix. Not only
is this vital, it often not covered in quantum mechanics
classes, both undergraduate and graduate.

It is also worth emphasizing that this chapter is primarily aimed at
physicists and for those others who are interested in the background
physics. However, it is not necessary for much of what follows.

===Schrodinger's Equation===

A common starting point in quantum mechanics is Schrodinger's equation. This equation is not derived or justified here, but is given in a general form:
{{Equation|<math>
H \left\vert \Psi\right\rangle = i\hbar\frac{\partial}{\partial t}\left\vert \Psi\right\rangle,
\,\!</math>|3.1}}<br />
where <math>H\,\!</math> is the Hamiltonian, 
<math>\hbar\,\!</math> is Planck's constant

(divided by <math>2\pi\,\!</math>), and <math>t\,\!</math> is time. The Hamiltonian contains what
is known about the system's evolution.
Most of the time in these notes, we let <math>\hbar = 1\,\!</math>.

This equation is (formally) solved by taking the time derivative to be
an ordinary derivative (we assume no explicit time dependence for
<math>H \,\!</math>), so
{{Equation|<math>
H \left\vert \Psi\right\rangle = i\frac{d \left\vert \Psi\right\rangle}{dt}.
\,\!</math>|3.2}}<br />
This means that
{{Equation|<math>
-iHdt = \frac{d \left\vert \Psi\right\rangle}{\left\vert \Psi\right\rangle},
\,\!</math>|3.3}}<br />
so
{{Equation|<math>\begin{align}
\ln \left\vert \Psi\right\rangle &= -iHt + C, \\
\Rightarrow\left\vert \Psi(t)\right\rangle &= e^{-iHt}\left\vert \Psi(0)\right\rangle.
\end{align}
\,\!</math>|3.4}}
Now if <math>H\,\!</math> is Hermitian (it is), then the matrix
{{Equation|<math>
U = e^{-iHt}
\,\!</math>|3.5}}<br />
is unitary. 
(If this is unclear, see [[Appendix C - Vectors and Linear Algebra]], in particular the section entitled [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|Unitary Matrices]].) Any
transformation on a closed system can be described by a unitary
transformation and any unitary transformation can be obtained by the
exponentiation of a Hermitian matrix.

The end result and important point is that the evolution of a quantum
state is, in general, given by a unitary matrix
{{Equation|<math>
\left\vert \Psi(t)\right\rangle = U\left\vert \Psi(0)\right\rangle.
\,\!</math>|3.6}}<br />
So our objective in quantum information processing is to create a
unitary evolution, and eventual measurement, which will produce a
particular outcome.

====Exponentiating a Matrix====

<div id="expmatrix"> ''Aside: a note about the exponentiation of a matrix.''</div>

It may seem strange to exponentiate a matrix. However, you can define
a function of a matrix according to its Taylor expansion. The details
of this are primarily unimportant here, but for demonstration purposes,
it is written out.

The Taylor expansion of an exponential is the following:
{{Equation|<math>
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}
\,\!</math>|3.7}}<br />
and this can be used to exponentiate a matrix by letting the matrix
replace <math>x\,\!</math> in the equation. This can also be used to prove that
{{Equation|<math>
e^{ix}=\cos x +i\sin x.
\,\!</math>|3.8}}<br />
''End Aside''

===Density Matrix for Pure States===

Now let us consider the object (a ''density matrix, or
density operator, of rank one'') 
{{Equation|<math>
\rho = \left\vert\psi\right\rangle \left\langle \psi\right\vert,
</math>|3.9}}<br />
which is just the outer product of two vectors. (See [[Appendix C - Vectors and Linear Algebra#Outer Product|Appendix C, Sec. C.2.4]].)

Since <math>\left\vert \psi\right\rangle = \left\vert \psi(t)\right\rangle\,\!</math>, <math>\rho=\rho(t)\,\!</math> is also true. If we
differentiate this with respect to <math>t\,\!</math>, we discover
{{Equation|<math>\begin{align}
\frac{\partial \rho }{\partial t} &=
\left(\frac{\partial \left\vert \psi\right\rangle}{\partial t}\right)\left\langle\psi\right\vert
+ \left\vert \psi\right\rangle\left(\frac{\partial \left\langle\psi\right\vert}{\partial t}\right)\\
&= (-iH)\rho + \rho (iH) = -i[H,\rho].
\end{align}
</math>|3.10}}
This is merely the Schrodinger equation for a density matrix with the solution
{{Equation|<math>
\rho(t) = U\rho(0)U^\dagger.
</math>|3.11}}<br />
This follows from <math>\left\vert\psi(t)\right\rangle\left\langle\psi(t)\right\vert =
U\left\vert\psi(0)\right\rangle\left\langle\psi(0)\right\vert U^\dagger\,\!</math>.

Consider our two-state system,
{{Equation|<math>
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 \end{array}\right),
\;\;\; \mbox{and} \;\;\;
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 \end{array}\right).
</math>|3.12}}<br />
Recall that the arbitrary superposition of these states is shown by
{{Equation|<math>
\left\vert \psi\right\rangle = \alpha_0\left\vert 0\right\rangle + \alpha_1\left\vert 1\right\rangle
= \left(\begin{array}{c} \alpha_0 \\ \alpha_1 \end{array}\right),
</math>|3.13}}<br />
where <math>\alpha_0\,\!</math> and <math>\alpha_1\,\!</math> are complex numbers such that
<math>|\alpha_0|^2 + |\alpha_1|^2 = 1\,\!</math>. The corresponding
''pure state'' (i.e. rank one) ''density matrix'' is given by
{{Equation|<math>
\rho_p = \left\vert\psi\right\rangle\left\langle\psi\right\vert
= \left(\begin{array}{cc}
|\alpha_0|^2 & \alpha_0 \alpha_1^* \\
\alpha_0^* \alpha_1 & |\alpha_1|^2 \end{array}\right).
</math>|3.14}}<br />
Note that the superposition in Eq.[[#eq3.13|(3.13)]] can be obtained from any pure state by a unitary transformation. Here, the trace of
the density matrix is an important quantity; it is
{{Equation|<math>
\mbox{Tr}(\rho_p) = |\alpha_0|^2 + |\alpha_1|^2 = 1.
\,\!</math>|3.15}}<br />
Notice also that the determinant of this matrix is zero, indicating that it has a zero eigenvalue:
{{Equation|<math>
\det(\rho_p) = |\alpha_0|^2|\alpha_1|^2 - \alpha_0 \alpha_1^*\alpha_0^*
\alpha_1 = 0.
</math>|3.16}}<br />
To see this another way, note that the density operator of rank one can be written as <math>U(\left\vert 0\right\rangle\left\langle0\right\vert)U^\dagger\,\!</math>, so that the determinant is
{{Equation|<math>\begin{align}
\det(U(\left\vert 0\right\rangle \left\langle 0\right\vert)U^\dagger) &= \det(U(\left\vert 0\right\rangle\left\langle 0\right\vert)U^{-1})\\
&= \det(U)\det(\left\vert0\right\rangle\left\langle 0\right\vert)\frac{1}{\det(U)} \\
&= \det(\left\vert 0\right\rangle \left\langle 0\right\vert) = 0.
\end{align}
</math>|3.17}}
This is a characteristic of a pure state and for two-state systems; it is a necessary and sufficient condition for the density operator to represent a pure state of the system.

===Measurements Revisited===

If the state of a quantum system is described by
{{Equation|<math>
\left\vert \psi\right\rangle = \alpha_0\left\vert 0\right\rangle + \alpha_1\left\vert 1\right\rangle,
</math>|3.18}}<br />
then the probability of finding it in the state <math>\left\vert 0\right\rangle\,\!</math> when measured in
the computational basis is <math>|\alpha_0|^2\,\!</math>. However, this is a
particular superposition that could be written as
{{Equation|<math>
\left\vert \psi\right\rangle = U \left\vert 0\right\rangle.
</math>|3.19}}<br />
In the section entitled [[#Schrodinger's Equation|Schrodinger's Equation]] it was shown that this matrix <math>U\,\!</math> results
from the exponentiation of a Hermitian matrix. Recall from the section entitled [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|The Pauli Matrices]] that any <math>2\times 2\,\!</math>
Hermitian matrix can be written in terms of the Pauli matrices. To make this explicit using standard conventions,
{{Equation|<math>\begin{align}
\left\vert \psi\right\rangle &= U\left\vert 0\right\rangle \\
&= \exp(-i\vec{n}\cdot\vec{\sigma} \theta) \left\vert 0\right\rangle \\
&= (\mathbb{I}\cos(\theta) -i\vec{n}\cdot\vec{\sigma} \sin(\theta))\left\vert 0\right\rangle,
\end{align}
</math>|3.20}}
where <math>\vec{n}\,\!</math> is a unit vector, <math>|\vec{n}|=1\,\!</math> and <math>\vec{n}\cdot\vec{\sigma} =
n_1\sigma_1+n_2\sigma_2+n_3\sigma_3\,\!</math>.
One can write this matrix out explicitly,
{{Equation|<math>\begin{align}
\exp(-i\vec{n}\cdot\vec{\sigma} \theta) &= \left(\begin{array}{cc}
1 & 0 \\
0 & 1 \end{array}\right)\cos(\theta) \\
& \;\;\; + (-i)\left[ n_1\left(\begin{array}{cc}
0 & 1 \\
1 & 0 \end{array}\right)
+ n_2\left(\begin{array}{cc}
0 & -i \\
i & 0 \end{array}\right)
+ n_3\left(\begin{array}{cc}
1 & 0 \\
0 & -1 \end{array}\right)\right]\sin(\theta) \\
&=
\left(\begin{array}{cc}
\cos(\theta) -in_3\sin(\theta) & (-in_1-n_2)\sin(\theta) \\
(-in_1+n_2)\sin(\theta) & \cos(\theta) +in_3\sin(\theta) \end{array}\right).
\end{align}
</math>|3.21}}
Notice this is a ''special unitary matrix.'' (See [[Appendix C - Vectors and Linear Algebra]], in particular the subsection [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|Unitary Matrices]].)

To see that any state <math>\left\vert \psi\right\rangle\,\!</math> for arbitrary coefficients
<math>\alpha_0\,\!</math>, <math>\alpha_1\,\!</math> can be obtained by choosing <math>\vec{n}\,\!</math> and <math>\theta\,\!</math>
appropriately, the state <math>\left\vert 0\right\rangle\,\!</math> can be chosen as a starting point.
Then,
{{Equation|<math>\begin{align}
U\left\vert 0\right\rangle &= \left(\begin{array}{cc}
\cos(\theta) -in_3\sin(\theta) & (-in_1-n_2)\sin(\theta) \\
(-in_1+n_2)\sin(\theta) & \cos(\theta) +in_3\sin(\theta)
\end{array}\right)
\left(\begin{array}{c} 1 \\ 0\end{array}\right) \\
&= \left(\begin{array}{c}
\cos(\theta) -in_3\sin(\theta) \\
(-in_1+n_2)\sin(\theta)\end{array}\right).
\end{align}
</math>|3.22}}
For example, choosing <math>\theta=0\,\!</math> gives the original state; choosing
<math>\vec{n} = (0,1,0)\,\!</math> and <math>\theta = \pi/2\,\!</math> gives <math>\left\vert 1\right\rangle\,\!</math>; and choosing
<math>\vec{n} = (0,1,0)\,\!</math> and <math>\theta = \pi/4\,\!</math> gives an equal superposition.
In general, when the system is in the state <math>\left\vert \psi\right\rangle = U\left\vert 0\right\rangle\,\!</math>,
the probability of finding the state <math>\left\vert 0 \right\rangle \,\!</math> when a measurement is made in the computational basis is given by
{{Equation|<math>\begin{align}
|\left\langle 0\right\vert U\left\vert 0\right\rangle|^2 &= |\cos(\theta) -in_3\sin(\theta)|^2 \\
&= \cos^2(\theta) +n_3^2\sin^2(\theta),
\end{align}
</math>|3.23}}
and the probability of finding <math>\left\vert 1\right\rangle\,\!</math> is
{{Equation|<math>\begin{align}
|\left\langle 1\right\vert U\left\vert 1\right\rangle|^2 &= |(-in_1+n_2)\sin(\theta)|^2 \\
&= (n_1^2+n_2^2)\sin^2(\theta).
\end{align}
</math>|3.24}}
Notice that the probabilities add up to one if <math>\vec{n}\,\!</math> is a unit vector.

What this shows is that there is a transformation that takes the state
<math>\left\vert 0\right\rangle\,\!</math>, which has probability <math>1\,\!</math> of being in the state <math>\left\vert 0\right\rangle\,\!</math> and
probability <math>0\,\!</math> of being in the state <math>\left\vert 1\right\rangle\,\!</math>, and transforms it
(using a <nowiki>"rotation''</nowiki>) into a state with a different (and generic)
probability of each. This means that the density matrix corresponding
to this system always has determinant zero, meaning that(for a two-state system) it has one
eigenvalue 1 and another eigenvalue 0. (The determinant is the
product of the eigenvalues.)

===Density Matrix for Mixed States===

For a system with <math>D\,\!</math> dimensions, a ''mixed state density matrix''
(or density operator, see Appendix \ref{app:cohvec})  is a matrix that is used to
describe a more general state of a quantum system. This can be written as
{{Equation|<math>
\rho_D = \sum_i a_i \rho_i
\,\! </math>|3.25}}<br />
where <math>a_i\geq 0\,\!</math>, <math>{\sum}_i a_i=1\,\!</math>, and the <math>\rho_i\,\!</math> are pure states. There is also a generalization of the Bloch sphere which is described in Appendix {app:polvec}.

Mixed state  density matrices are important in all descriptions of physical implementations of quantum information processing. For this reason, a bit of labor should go into understanding the density matrix. The rest of this section is devoted to the physical interpretation and properties of this description of a quantum system. The first description presented is called the ensemble interpretation of the density matrix. This is perhaps the easiest to understand. Another set of physical systems that are described by density matrices will be given elsewhere.

====General Properties====

In general, a density matrix has the following properties:
{{Equation|<math>\begin{align}
\rho = \rho^\dagger, &\;\;\; \mbox{it is hermitian}, \\
\rho \geq 0,\; &\;\;\; \mbox{it is positive semi-definite},
\\
\mbox{Tr}(\rho) = 1,\; &\;\;\; \mbox{it is normalized}.
\end{align}
\,\!</math>|3.26}}
If, in addition, it is a pure state, then
{{Equation|<math>
\rho^2 = \rho.
\,\!</math>|3.27}}<br />
The second property in Eq.[[#eq3.28|(3.28)]] really means that the eigenvalues of the density matrix are greater than or equal to zero.

====Density Matrix for a Mixed State: Two States====

A mixed state density matrix for a two-state system is a rank two density matrix, <math>\rho_m\,\!</math>, which can be described by
{{Equation|<math>
\rho_m = \left[a_1\rho_1 + a_2\rho_2\right],
</math>|3.28}}<br />
where <math>\rho_1 = \left\vert\psi_1\right\rangle\left\langle \psi_1\right\vert\,\!</math>, <math>\rho_2 = \left \vert \psi_2\right\rangle\left\langle \psi_2\right\vert \,\!</math>
and <math>a_1 + a_2=1\,\!</math>. The <math>a_i\,\!</math> are probabilities and must sum to one.
(Note, if <math>\left\vert \psi\right\rangle_1=\left\vert \psi\right\rangle_2\,\!</math>, or if one <math>a_i\,\!</math> or one
<math>\left\vert \psi\right\rangle_i\,\!</math> is zero, then this reduces to a pure state.) In this mixture,
the probability of finding the state <math>\left\vert \psi_1\right\rangle\,\!</math> is <math>a_1\,\!</math>
and the probability of finding the state <math>\left\vert \psi_2\right\rangle\,\!</math> is <math>a_2\,\!</math>.

====Description of Open Quantum Systems: An Example====

One example of the utility of a density matrix is the following
statistical problem. Let us consider a collection of electrons in a box, where their
spin is a two-state system being either up or down when measured. If
a subset of these electrons is prepared in the state <nowiki>''up''</nowiki> before
being put in the box and the rest <nowiki>''down,''</nowiki> then the description of
the system of particles is given by
{{Equation|<math>
\rho = a_u \left\vert\uparrow\right\rangle\left\langle\uparrow\right\vert +
a_d\left\vert\downarrow\right\rangle\left\langle\downarrow\right\vert,
</math>|3.29}}<br />
where the fraction of <nowiki>''up''</nowiki> particles is <math>a_u\,\!</math> and the fraction of <nowiki>''down''</nowiki> is <math>a_d\,\!</math>. Our system is described by this density matrix---if a particle is chosen at random from the box and measured, the state of the particle is <math>\left\vert \uparrow\right\rangle\,\!</math> with probability <math>a_u\,\!</math>
and <math>\left\vert \downarrow\right\rangle\,\!</math> with probability <math>a_d\,\!</math>. This is known as the "statistical
interpretation" of the density operator.

There is another example that is more relevant for our purposes. Let us consider another two-state system.
If there is some probability <math>p\,\!</math> for an error to occur, let us say it is a unitary operator <math>U_e\,\!</math>, then the density matrix for the
system is
{{Equation|<math>
\rho_e = (1-p)\left\vert\psi\right\rangle\left\langle\psi\right\vert + pU_e\left\vert\psi\right\rangle\left\langle\psi\right\vert U_e^\dagger.
</math>|3.30}}<br />
This is the same form as Eq.[[#eq3.31|(3.31)]].

Note that in each
case the probabilities associated with the density matrix <math>p,1-p\,\!</math>, and
<math>a_u,a_d\,\!</math>, (generally, <math>a_i\,\!</math>) are classical probabilities;
they are associated with a classical probability distribution---the
probability for error/no error and up/down. These are not
probabilities associated with the superposition of the quantum state
in the equation <math>\left\vert \psi\right\rangle = \alpha_0 \left\vert 0\right\rangle + \alpha_1\left\vert 1\right\rangle\,\!</math>
given by the square of the moduli of the coefficients. This is an
important distinction! The state
<math>\left\vert \psi\right\rangle\,\!</math> can be taken to the state <math>\left\vert 0\right\rangle\,\!</math> with a unitary
transformation. This state is deterministic in the sense that the
result <math>\left\vert 0\right\rangle\,\!</math> will be obtained from a measurement in the
computational basis since there is no probability for obtaining
<math>\left\vert 1\right\rangle\,\!</math>. However, for nonzero <math>\left\vert \psi\right\rangle\,\!</math> and a non-identity
operator <math>U\,\!</math>, the matrix <math>\rho_e\,\!</math> has rank two and thus can never have
probability <math>1\,\!</math> for either of the two states, <math>\left\vert 0\right\rangle\,\!</math> or <math>\left\vert 1\right\rangle\,\!</math>.
Thus we have maximum knowledge about a pure state since
there is a way to choose a measurement, perhaps after a unitary
transformation, which achieves a certain result with probability <math>1\,\!</math>.
For the mixed state density operator this is not possible. The state
{{Equation|<math>
\rho = \left(\begin{array}{cc}
1/2 & 0 \\
0 & 1/2 \end{array}\right),
</math>|3.31}}<br />
for which we have the least amount of knowledge, is called the
maximally mixed state.  The
state could be either up or down with equal probability and neither is
a better guess. If the two eigenvalues are not equal, then there is a
better guess (or bet) as to the result of a measurement. If one
eigenvalue is zero, then there is a definite best guess.

To be more specific, independent of basis (unitary transformations),
one always has a probability greater than zero of measuring
<math>\left\vert \uparrow\right\rangle\,\!</math> and probability greater than zero of measuring
<math>\left\vert \downarrow\right\rangle\,\!</math>. Thus the state described by the density matrix is
a ''mixed state''  in the sense
that it can be considered a statistical mixture of the two states
<math>\left\vert \uparrow\right\rangle\,\!</math> and <math>\left\vert \downarrow\right\rangle\,\!</math>. This, because classical
probabilities are included separately, is significantly different from
the pure state density matrix, which is a special case of all density
matrices.

To see that mixtures remain after a unitary transformation on the
system, note that a unitary matrix does not change the eigenvalues.
This is because the eigenvalue equation is the same for a Hermitian
matrix and its corresponding diagonal matrix. Let <math>\rho =
U\rho_d U^\dagger\,\!</math>. It can now be seen,
{{Equation|<math>\begin{align}
\det(\rho -\lambda\mathbb{I}) &=
\det(U(\rho_d-\lambda\mathbb{I})U^\dagger) \\
&=
\det(U)\det(\rho_d-\lambda\mathbb{I})\det(U^\dagger) \\
&=
\det(U)\det(\rho_d-\lambda\mathbb{I})\det(U^{-1}) \\
&= \det(\rho_d-\lambda\mathbb{I}).
\end{align}
</math>|3.32}}

====Two-State Example: Bloch Sphere====

Since our interest is primarily in qubits, which are two-state
systems, we return to a two-state example.

A very convenient representation of two state density matrices, one
that can written in the so-called Bloch sphere 
representation given the fact that the density matrix is Hermitian,
{{Equation|<math>
\rho_2 = \frac{1}{2}(\mathbb{I} + \vec{n}\cdot\vec{\sigma}),
</math>|3.33}}<br />
where, for the density matrix to be positive <math>|\vec{n}| \leq 1\,\!</math>, and the
<math>\sigma_i\,\!</math> are the Pauli matrices 
{{Equation|<math>
\vec{\sigma} = (\sigma_x,\sigma_y,\sigma_z) = \left(
\left(\begin{array}{cc}
0 & 1 \\
1 & 0 \end{array}\right),
\left(\begin{array}{cc}
0 & -i \\
i & 0 \end{array}\right),
\left(\begin{array}{cc}
1 & 0 \\
0 & -1 \end{array}\right)
\right).
</math>|3.34}}<br />
The matrix entries on the RHS of this equation are the [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|The Pauli matrices]] discussed above. It is not difficult to convince yourself that any Hermitian matrix can be written as a real linear combination of the three Pauli matrices and the identity. The eigenvalues are given by
{{Equation|<math>
\lambda_\pm = \frac{1\pm|\vec{n}|}{2}.
</math>|3.35}}<br />
When <math>|\vec{n}| = 1\,\!</math>, the state is pure, i.e., that the matrix
has rank one since it has one eigenvalue one and one zero. If <math>|\vec{n}|
< 1\,\!</math>, the density matrix represents a mixed state since rank is
greater than one--there are two non-zero eigenvalues. These leads to
the following picture: the pure states lie on the surface of the
sphere (<math>\vec{n}\cdot \vec{n} =1\,\!</math>), and mixed states lie in the interior of
the sphere with the maximally mixed state at the origin. This is
supposedly due to Bloch. Hence the name Bloch sphere.

Using <math>\rho^2 =\rho\,\!</math> the condition that <math>\vec{n}\cdot\vec{n} =1\,\!</math> for a pure
state can also be determined. The square in the Bloch sphere
representation yields
{{Equation|<math>
\rho_2^2 = \frac{1}{4}\left(\mathbb{I} + 2\vec{n}\cdot\vec{\sigma} + (\vec{n}\cdot\vec{\sigma})^2\right),
</math>|3.36}}<br />
and using
{{Equation|<math>
\sigma_i \sigma_j = \mathbb{I}\delta_{ij} + i\epsilon_{ijk}\sigma_k,
</math>|3.37}}<br />
then <math>\rho_2^2 =\rho_2\,\!</math> if and only if <math>\vec{n}\cdot\vec{n} =1\,\!</math>. This technique is
used for higher dimensions. See [[Appendix E - Density Operator: Extensions|Appendix E]].

Two density matrices <math>\rho_1=(1/2)(\mathbb{I} +\vec{n}\cdot\vec{\sigma})\,\!</math> and
<math>\rho_2=(1/2)(\mathbb{I} +\vec{m}\cdot\vec{\sigma})\,\!</math>, correspond to orthogonal
states when
{{Equation|<math>\begin{align}
\mbox{Tr}(\rho_1\rho_2) &= \frac{1}{4}\mbox{Tr}\big(\mathbb{I} + (\vec{n}\cdot\vec{\sigma})(\vec{m}\cdot\vec{\sigma})\big) \\
&= \frac{1}{2}(1+\vec{n}\cdot\vec{m}) =0.
\end{align}</math>|3.38}}<br />
This implies that
{{Equation|<math>
\vec{n}\cdot\vec{m} = |\vec{n}||\vec{m}|\cos(\theta) = -1.
</math>|3.39}}<br />
Since the magnitudes must be one, the orthogonal states correspond to
pure states on a surface of a sphere which are represented by
antipodal points.

====Rotations of Bloch Vectors====

As shown above, the solution to the Schrodinger equation for the density operator is (see [[#eq3.11|Eq.(3.11)]])
<center><math>
\rho(t) = U(t)\rho(0) U^\dagger(t).
\,\!</math></center>
In general an open system will evolve according to
<center><math>
\rho = U \rho_0 U^\dagger,
\,\!</math></center>
whether or not the time dependence is explicitly taken into account. When the density operator is represented using the Bloch vector, the vector is rotated by the unitary transformation. This is seen through an explicit calculation.

There are two ways to see this. One is to simply act with the matrices in the Euler angle parameterization in [[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit|Section C.5.1]] one each of the Pauli matrices to show that indeed,
{{Equation|<math>
U\vec{m}\cdot\vec{\sigma} U^\dagger = \sum_{ij} m_i R_{ij} \sigma_j.
\,\!</math>|3.40}}
This is easily seen to be a standard rotation matrix. (See for example http://en.wikipedia.org/wiki/Rotation_matrix.)

Another way to do this is to take
<center><math>
\rho_0 = \frac{1}{2}(\mathbb{I} + \vec{m}\cdot\vec{\sigma}),
\,\!</math></center>
as in [[#eq3.33|Eq.(3.33)]]. (Recall <math>\vec{m}\cdot\vec{\sigma} = \sum\!{}_i \; m_i \sigma_i\,\!</math>.) Now act on <math>\rho</math> with <math>U\,\!</math> as given in [[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit|Section C.5.1]] by the so-called adjoint action <math>\rho = U \rho_0 U^\dagger\,\!</math>,
{{Equation|<math>
\rho = [\mathbb{I}\cos(\theta/2)-i\vec{n}\cdot\vec{\sigma}\sin(\theta/2)]\frac{1}{2}(\mathbb{I} + \vec{m}\cdot\vec{\sigma})[\mathbb{I}\cos(\theta/2)+i\vec{n}\cdot\vec{\sigma}\sin(\theta/2)].
\,\!</math>|3.41}}
To do this calculation explicitly, it helps (but is not necessary) to use the following identity,
{{Equation|<math>
\sum_i \epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}.
\,\!</math>|3.42}}
Then, if one only considers the non-trivial part of the density operator, <math>\vec{m}\cdot\vec{\sigma}\,\!</math>, the result is
{{Equation|<math>
e^{-i\vec{n}\cdot\vec{\sigma}\theta/2} \vec{m}\cdot\vec{\sigma} e^{i\vec{n}\cdot\vec{\sigma}\theta/2}
= \vec{m}\cdot\vec{\sigma} \cos(\theta) + (\vec{n}\cdot\vec{m}) (\vec{n}\cdot\vec{\sigma})(1-\cos(\theta)) + (\vec{n}\times \vec{m})\cdot\vec{\sigma}\sin(\theta),
\,\!</math>|3.43}}
or
{{Equation|<math>\begin{align}
e^{-i\vec{n}\cdot\vec{\sigma}\theta/2} \vec{m}\cdot\vec{\sigma} e^{i\vec{n}\cdot\vec{\sigma}\theta/2}
&= \frac{1}{2}\vec{m}\cdot\vec{\sigma} \cos(\theta) + \frac{1}{2}(\vec{n}\cdot\vec{m}) (\vec{n}\cdot\vec{\sigma})\cos(\theta) + (\vec{n}\cdot\vec{m}) (\vec{n}\cdot\vec{\sigma}) \\
& \;\;\;\; + (\vec{n}\times \vec{m})\cdot\vec{\sigma}\sin(\theta)
+\frac{1}{2}[(\vec{n}\times\vec{m})\times\vec{n}]\cdot\vec{\sigma}\cos(\theta)\end{align}
\,\!</math>|3.44}}
where
{{Equation|<math>
(\vec{n}\times \vec{m})\cdot\vec{\sigma} = \sum_{ijk} \epsilon_{ijk} n_im_j\sigma_k.
\,\!</math>|3.45}}
Therefore, the result of the action of <math>U\,\!</math> is to produce, from <math>\vec{m}\,\!</math>, the vector
{{Equation|<math>
\vec{m}^\prime= \vec{m} \cos(\theta) + (\vec{n}\cdot\vec{m})\vec{n}(1-\cos(\theta)) + (\vec{n}\times \vec{m})\sin(\theta).
\,\!</math>|3.46}}
This equation can be interpreted as follows. We consider three components of the vector, the part along the axis of rotation and the two parts in the plane perpendicular to the axis of rotation. The part of the vector along the axis of rotation <math>(\vec{m}\cdot\vec{n})\vec{n}\,\!</math> does not change. The parts perpendicular to <math>(\vec{m}\cdot\vec{n})\vec{n}\,\!</math> change just like a vector rotated in a plane, but these parts are rotated in the plane perpendicular to the rotation axis and sitting at the end of the vector <math>(\vec{m}\cdot\vec{n})\vec{n}\,\!</math>. It takes a bit of geometry and vector algebra to show this is the case.

===Expectation Values===

The expectation value 
of an operator <math>\mathcal{O}\,\!</math>, is given by
{{Equation|<math>
\langle \mathcal{O} \rangle = \mbox{Tr}(\rho \mathcal{O}),
</math>|3.47}}<br />
and is the "average value" of the operator. For a pure state
<math>\rho_p = \left\vert\psi\right\rangle\left\langle\psi\right\vert\,\!</math>, this reduces to
{{Equation|<math>
(\langle \mathcal{O} \rangle)_p = \left\langle\psi\right\vert \mathcal{O}\left\vert \psi\right\rangle.
</math>|3.48}}<br />

[[Chapter 4 - Entanglement#Introduction|Continue to '''Chapter 4 - Entanglement''']]

Appendix D - Group Theory

2011-11-01T03:54:45Z

Stempel2: /* Concluding Remarks */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.17}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.18}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.19}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.18|Equation (D.18)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices that represent generators of <math>SU(d)\,\!</math>:
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.20}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.21}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra, we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,
\,\!</math>|D.22}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.23}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.24}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

Also provided in cite{Macfarlane} are the following identities:
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.25}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.26}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.27}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation---this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, we need to find the set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients. These not only put the tensor product of the vectors into this special form, but they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets of matrices that represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices are all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T03:49:19Z

Stempel2: /* Addition of Angular Momenta */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.17}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.18}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.19}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.18|Equation (D.18)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices that represent generators of <math>SU(d)\,\!</math>:
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.20}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.21}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra, we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,
\,\!</math>|D.22}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.23}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.24}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

Also provided in cite{Macfarlane} are the following identities:
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.25}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.26}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.27}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation---this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, we need to find the set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients. These not only put the tensor product of the vectors into this special form, but they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T03:12:53Z

Stempel2: /* Tensor Products of Representations */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.17}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.18}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.19}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.18|Equation (D.18)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices that represent generators of <math>SU(d)\,\!</math>:
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.20}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.21}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra, we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,
\,\!</math>|D.22}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.23}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.24}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

Also provided in cite{Macfarlane} are the following identities:
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.25}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.26}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.27}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation---this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T03:09:59Z

Stempel2: /* Some Useful Relations Among Lie Algebra Elements */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.17}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.18}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.19}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.18|Equation (D.18)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices that represent generators of <math>SU(d)\,\!</math>:
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.20}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.21}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra, we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,
\,\!</math>|D.22}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.23}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.24}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

Also provided in cite{Macfarlane} are the following identities:
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.25}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.26}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.27}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T03:03:37Z

Stempel2: /* Representation Theory for Lie Groups */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.17}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.18}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.19}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T03:03:10Z

Stempel2: /* The Lie Algebra of a Lie Group */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.17}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T03:00:35Z

Stempel2: /* Representation Theory for Lie Groups */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices that are reducible to block-diagonal form with blocks that cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set that obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation, although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T02:22:38Z

Stempel2: /* The Lie Algebra of a Lie Group */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group that is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose that there is a Lie algebra corresponding to a Lie group that has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group written in terms of these parameters is
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group, and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. (For example, subalgebras correspond to subgroups.)

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T02:14:45Z

Stempel2: /* More Representation Theory */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In physics we are most often concerned with linear representations of groups that use linear operators to represent group elements; we represent these operators with matrices. In this Appendix, the focus is entirely on these types of representations, although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks make up "irreducible representations." Our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly, a set of matrices that may be block-diagonalizable but has been acted upon by a highly non-trivial <math>S \,\!</math> may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T02:05:20Z

Stempel2: /* Example 9 */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing, an important set of unitary groups is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T02:01:52Z

Stempel2: /* Example 8 */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex <math>2\times 2\,\!</math> matrices
that satisfy
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math> and is the set of ''unitary''
<math>2\times 2\,\!</math> matrices (hence the <math>U\,\!</math>). Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math> where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group that is often considered---the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>
and is known as the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math>, and the "S" for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:52:27Z

Stempel2: /* Example 7 */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math> for all
<math>\theta\,\!</math>. This group has an infinite number of elements (i.e. an infinite order) and one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold---a circle. Notice that this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle; after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:45:38Z

Stempel2: /* Infinite Order Groups: Lie Groups */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups that have infinite order can be described with one or more parameters. Groups that are differentiable with respect to those parameters are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group that is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:32:53Z

Stempel2: /* Definition 16: Coset */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup and <math>G\,\!</math> be an element of the group <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that they partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. You could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:29:35Z

Stempel2: /* Definition 15: Normalizer */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:28:33Z

Stempel2: /* Definition 14: Generators of a Group */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:25:16Z

Stempel2: /* Definition 13: Weight of an Operator */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. Its importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:24:44Z

Stempel2: /* Properties of the Pauli Group */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits with the tensor product symbols omitted. The following are elements:
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math>, of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2.
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:20:54Z

Stempel2: /* Definition 10: Stabilizer */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>m\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> that leaves the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or sometimes in physics, '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:09:28Z

Stempel2: /* Equivalent Representations */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math> such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math> called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:07:27Z

Stempel2: /* Example 6 Continued */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

Example 6 is a non-trivial problem even though it appears
otherwise. The way to show this is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math>, on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices that are known to form a representation of the group, it is non-trivial to find the similarity transformation that will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations, <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math>, such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>, otherwise they are inequivalent.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math>, called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:05:12Z

Stempel2: /* Definition 9: Similarity Transformation */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math> that looks like
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>, then
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion on similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

However, Example 6 is a non-trivial problem even though the problem appears
trivial. The way to show that it is non-trivial is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math> on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices which are known to form a representation of the group, it is non-trivial to find the similarity transformation which will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations, <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math>, such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>, otherwise they are inequivalent.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math>, called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T01:02:04Z

Stempel2: /* Example 6 */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix, and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math>
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}.\,\!</math>, <math>B^\prime = SBS^{-1}.\,\!</math>, and <math>C^\prime = SCS^{-1}.\,\!</math>,
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion about similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

However, Example 6 is a non-trivial problem even though the problem appears
trivial. The way to show that it is non-trivial is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math> on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices which are known to form a representation of the group, it is non-trivial to find the similarity transformation which will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations, <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math>, such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>, otherwise they are inequivalent.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math>, called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T00:55:11Z

Stempel2: /* Example 6 */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations that will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices:
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly, when these matrices act on a column vector, labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right),
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different; in fact, the dimensions of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above,
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>), since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> that is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix, and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math>
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}.\,\!</math>, <math>B^\prime = SBS^{-1}.\,\!</math>, and <math>C^\prime = SCS^{-1}.\,\!</math>,
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion about similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

However, Example 6 is a non-trivial problem even though the problem appears
trivial. The way to show that it is non-trivial is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math> on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices which are known to form a representation of the group, it is non-trivial to find the similarity transformation which will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations, <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math>, such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>, otherwise they are inequivalent.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math>, called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-11-01T00:46:51Z

Stempel2: /* Definition 8: Representation */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
This, along with ordinary
matrix multiplication for the product rule, provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations which will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices.
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly when these matrices act on a column vector labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right).
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different. In fact the dimension of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>) since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> which is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix, and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math>
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}.\,\!</math>, <math>B^\prime = SBS^{-1}.\,\!</math>, and <math>C^\prime = SCS^{-1}.\,\!</math>,
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion about similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

However, Example 6 is a non-trivial problem even though the problem appears
trivial. The way to show that it is non-trivial is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math> on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices which are known to form a representation of the group, it is non-trivial to find the similarity transformation which will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations, <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math>, such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>, otherwise they are inequivalent.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math>, called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-10-31T18:20:04Z

Stempel2: /* Definition 8: Representation */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
Then, along with ordinary
matrix multiplication for the product rule, this provides a way to represent
any group. This is true for groups that have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' that can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity and the multiplication relations (in the group multiplication table) are preserved; this representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices that can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations which will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices.
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly when these matrices act on a column vector labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right).
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different. In fact the dimension of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>) since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> which is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix, and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math>
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}.\,\!</math>, <math>B^\prime = SBS^{-1}.\,\!</math>, and <math>C^\prime = SCS^{-1}.\,\!</math>,
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion about similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

However, Example 6 is a non-trivial problem even though the problem appears
trivial. The way to show that it is non-trivial is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math> on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices which are known to form a representation of the group, it is non-trivial to find the similarity transformation which will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations, <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math>, such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>, otherwise they are inequivalent.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math>, called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.

Appendix D - Group Theory

2011-10-31T18:16:05Z

Stempel2: /* The Rearrangement Theorem */

===Introduction===

<nowiki>''</nowiki>''Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.''<nowiki>''</nowiki>

'''Hermann Weyl'''

====Symmetries and Groups====

Symmetry arguments have been used widely in mathematics, physics,
chemistry, biology, computer science, engineering, and elsewhere.
Group theory can be an invaluable organizational tool,
whether it is used explicitly or implicitly, in many areas of
science.

In physics, symmetry principles are often used to describe what
changes and what does not in a physical system undergoing some
particular transformation. For example, if a knob is turned in an
experiment and nothing changes, then that is an invariant of the
system and thus indicates a symmetry. (Of course, the trivial case
where the knob has nothing to do with the experiment, like if the
machine with the knob is unplugged, should be excluded.) The objective
here is to explain group theory with this practical viewpoint in
mind; the idea is for this motivation to be kept in mind
throughout these notes. Some formalism is necessary however.

It is worth noting that very general things tend to need to be
abstract. And so it is with group theory. However, to reiterate, the
objective here is to be as concrete as possible with the emphasis on
physical applications. In this regard, it is worth mentioning that,
directly or indirectly, [[Bibliography#Tinkham:gpthbook|Michael Tinkham's book]]
on group theory very much influenced these notes. Also, Encyclopedia of Maths, Hammermesh, ...

====Group Theory in Physics====

The applications to physics are too numerous to mention here. However, several comments
are in order. First, if a system has a symmetry (often able to be determined by inspection), then it has a
constraint placed on it. This limits the acceptability of solutions to a problem -
they must satisfy the symmetry requirement. Thus identifying
symmetries is an excellent problem-solving technique. Choosing
coordinates is an example of such symmetry identification.

A group is a set of symmetries. To see this, suppose that, for example, elements <math>A,B,C,D,...\,\!</math> operate on an object in such a way
that they do not change the object. Most often in physics the
elements are matrices and the objects on which they act are vectors.
If a vector or set of vectors is unchanged by these operations, then
the vectors have a symmetry described by the action of these
operators. In Example 2 the vectors are the vertices of the triangle
and the triangle is unchanged by the action of the group elements given in the example. (Notice, as an example
of how a set of symmetries forms a group, that if the vector is <math>v\,\!</math>
and assuming <math>Av=v\,\!</math>, i.e. <math>A\,\!</math> is a symmetry operation, and also assuming
<math>Bv=v\,\!</math>, then <math>ABv = v\,\!</math>. Thus the set is closed under multiplication.)
One way to think of this is quite literal. If a symmetry operation is
applied to the equilateral triangle and the triangle is still
equilateral with the vertices indistinguishable (assuming no labels), then the
operation did not change anything discernible.

It turns out that group theory has been applied with great success to
many areas of quantum physics: solid-state physics including
crystallography, nuclear physics, atomic physics, molecular physics,
and particle physics. It has also been applied in classical physics
and relativity. It has been especially indispensable in quantum field theory and particle physics where symmetries correspond to conserved quantities observed in experiment.

Some groups of infinite order, such as Lie groups, were originally
studied largely in order to understand the symmetries of
differential equations. This is the set of groups that is discussed
next.

===Definitions and Examples===

====Definition 1: Group====
A '''group''' <math>\mathcal{G}\,\!</math> is a set of objects <math>\{A,B,C,
...\}\,\!</math> together with a composition rule between them (denoted <math>\circ\,\!</math> here and
called a product or multiplication) such that the following are satisfied:
#<math>(A\circ B)\circ C = A\circ (B \circ C)\,\!</math>. (<math>\circ\,\!</math> is associative.)
#If <math>A\in \mathcal{G}\,\!</math> and <math>B\in\mathcal{G}\,\!</math>, then their product is <math>A\circ B \in \mathcal{G}\,\!</math>. (The set is closed under multiplication.)
#There is an element <math>\mathbb{I}\in \mathcal{G}\,\!</math> such that, for all <math>A\in \mathcal{G}\,\!</math>, <math>\mathbb{I}A = A = A\mathbb{I}\,\!</math>. (<math>\mathcal{G}\,\!</math> contains the identity element.)
#For all <math>A\in \mathcal{G}\,\!</math> there exists an element <math>A^{-1}\in\mathcal{G}\,\!</math> such that <math>AA^{-1} = \mathbb{I} =A^{-1}A\,\!</math>.

In the provided examples, the objective is to make the direct connection between a group and a set of symmetries of an object. The reason is that ''a set of symmetries forms a group'' since it satisfies all the conditions in the definition. The symmetries are things you can do to (i.e., operations you can perform on) a set that leaves the set unchanged.

To see this, suppose that we operate on a set of vectors whose endpoints have a certain symmetry associated with them. (For example, the vertices of a triangle.) Assume their origin is the origin of a coordinate axes. Operating on these with a set of matrix operators may leave the set of vectors unchanged, e.g. the arrows associated with the vectors still point to the same set of points, if the set of matrices is chosen properly. Assuming all possible such matrices are included in the set, then the set of matrices, or set of symmetries, forms a group.

====Example 1====

Consider a line segment of length 2 cm with midpoint at zero. Suppose the end points are located at <math>\pm 1\,\!</math> cm of the x-axis. If the line segment were rotated <math>180^o\,\!</math> about any line perpendicular to the segment, it would look like the same line segment. (Let us be definite and choose the axis perpendicular to the x-y plane after choosing x and y axes.) What this would do is exchange the two ends. The set of points <math>\pm 1\,\!</math> could be acted upon by an operator that
exchanges the two. This rotation operation can be represented through multiplication by <math>-1\,\!</math>. Then there are two elements in the set of operations to consider. The first is ''do nothing'' represented by <math>1\,\!</math>. (This, of course, is the identity operation <math>\mathbb{I}\,\!</math> for this ''group''.) The other element is <math>-1\,\!</math>. Thus, representing
multiplication by <math>\circ\,\!</math>, we have a group with the set <math>\{+1,-1\}\,\!</math>
and operation <math>\circ \equiv \times \,\!</math>. Clearly the product is associative (it is multiplication), the set contains the identity, products are either <math>+1\,\!</math> or <math>-1\,\!</math> which are both in the group (indicating closure), and the inverse of <math>-1\,\!</math> is <math>-1\,\!</math>; all of requirements defined above are satisfied. In fact this is the simplest group.

====Example 2====

The set of symmetries of an equilateral triangle can be represented in several ways. Two that are presented here are the set of operations on vectors from the origin to the vertices and the set of permutations on three objects.

<center> <div id="Figure D.1">'''Figure D.1'''</div>
{|
|[[File:triangle2.jpg]]
|}
Figure D.1: An equilateral triangle with vertices in the x-y plane, <math> v_1\,\!</math> at <math>(0,1/\sqrt{3})\,\!</math>, <math>v_2\,\!</math> at <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>, and <math>v_3\,\!</math> at <math>(1/2,-1/(2\sqrt{3}))\,\!</math>.
</center>

Consider an equilateral triangle with its
center at the origin of the x-y plane and vertices
placed at the following points: <math>(0,1/\sqrt{3})\,\!</math>,
<math>(1/2,-1/(2\sqrt{3}))\,\!</math>, <math>(-1/2,-1/(2\sqrt{3}))\,\!</math>. (See [[#Figure D.1|Figure D.1]].) Now consider the
following operations on the triangle: a rotation of <math>0^o\,\!</math> (do
nothing), a rotation of <math>120^o\,\!</math>, a rotation of <math>240^o\,\!</math>, and a reflection
about the <math>x\,\!</math> axis, <math>\sigma_1\,\!</math>. There are two other reflections we could perform, labelled <math>\sigma_2\,\!</math> and <math>\sigma_3\,\!</math>, which are reflections through lines bisecting the angles at vertices 2 and 3 respectively, as shown in [[#Figure D.1|Figure D.1]]. These make up the set of six symmetry operations on the equilateral triangle.

If we take the first of these, <math>P_0\,\!</math>, to be the original configuration (shown in [[#Figure D.1|Figure D.1]]), then each of the first three of these are a rotation from the original configuration. Each of the last three is obtained from a reflection combined with a rotation. To be explicit, let us consider the following operations:
{{Equation|<math>
\mathbb{I}_2 = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1
\end{array}\right),
R_1 = \left(\begin{array}{cc} -1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
R_2 = \left(\begin{array}{cc} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right),
\,\!</math>|D.1}}
where <math>R_1\,\!</math> is a rotation of the x-y plane by <math>120^o\,\!</math>, <math>R_2\,\!</math> is a rotation
of the x-y plane by <math>240^o\,\!</math>, and <math>\mathbb{I}_2\,\!</math> is a rotation of <math>0^o\,\!</math>. In addition to these operations, two others must be included
to complete the set:
{{Equation|<math>
\sigma_1 = \left(\begin{array}{cc} -1 &0 \\ 0 &1 \end{array}\right), \;\;
\sigma_2 =\sigma_1 R_1 = \left(\begin{array}{cc} 1/2 & \sqrt{3}/2 \\ \sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\sigma_3 = \sigma_1R_2 = \left(\begin{array}{cc} 1/2 & -\sqrt{3}/2 \\ -\sqrt{3}/2
& -1/2 \end{array}\right), \; \;
\,\!</math>|D.2}}
where <math>\sigma_1 R_1\,\!</math> is the same as <math>\sigma_1\circ R_1\,\!</math>, but the <math>\circ\,\!</math> has been dropped
since this is ordinary matrix multiplication. This group will be used
as an example for several group properties and is called <math>S_3\,\!</math>. The
products of these
elements are summarized in [[#Table D.1|Table D.1]], which is called the
multiplication table for the group. The multiplication table will be
discussed repeatedly throughout this appendix due to its importance in
group theory. It would be advisable to stare at it for some time to
see what patterns can be identified. The meaning of these patterns
will be discussed later.

<center><div id="Table D.1">
'''Table D.1: Group Multiplication Table for''' <math>S_3\,\!</math></div>
{| border="1" cellpadding="20" cellspacing="0"
|-
|align="center"|<math> \downarrow\rightarrow\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math>\mathbb{I}_2 \,\!</math>
|align="center"|<math>R_1 \,\!</math>
|align="center"|<math>R_2 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math>\sigma_2 \,\!</math>
|align="center"|<math>\sigma_3 \,\!</math>
|-
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|-
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> R_2\,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|-
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|align="center"|<math> R_1\,\!</math>
|align="center"|<math> R_2\,\!</math>
|-
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_2 \,\!</math>
|align="center"|<math> \sigma_3\,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2 \,\!</math>
|align="center"|<math> R_1 \,\!</math>
|-
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_3 \,\!</math>
|align="center"|<math> \sigma_1 \,\!</math>
|align="center"|<math> \sigma_2\,\!</math>
|align="center"|<math> R_1 \,\!</math>
|align="center"|<math> R_2 \,\!</math>
|align="center"|<math> \mathbb{I}_2\,\!</math>
|}
</center>
<center>
Table D.1: ''Group multiplication table for the group <math>S_3\,\!</math>. The notation in the upper left corner (<math>\downarrow\rightarrow\,\!</math>) indicates that the element in the first column is to be multiplied by the element in the first row to obtain the result. Since the group is not abelian, i.e. the elements do not commute, the order matters.''
</center>

A second way to identify all possible configurations of
the triangle that leave the triangle looking the same is to use the positions of the vertices. There are six possible choices for the positions of the vertices. Let us name them 1,2,3. Then, reading counter-clockwise
from the top, we can have <math>P_0=(1,2,3)\,\!</math>, <math>P_2=(3,1,2)\,\!</math>, <math>P_4=(2,3,1)\,\!</math>,
<math>P_1=(1,3,2)\,\!</math>, <math>P_3=(3,2,1)\,\!</math>, <math>P_5=(2,1,3)\,\!</math>. These are all of the permutations of three objects. (In this case the three objects are the numbers 1,2,3.) This is another way to represent the various configurations of the equilateral triangle.

====Definition 2: Order of a Group====

The number of elements in a group is called the '''order''' of the group.

Example 1 has two elements and so has order two. Example 2 has six
elements, so the order of this group is six.

====Definition 3: Abelian and Nonabelian Group====

A group for which every element of the group commutes with every other element of the group (<math>g_1g_2 = g_2g_1,\;\;\forall g_1,g_2\in \mathcal{G}\,\!</math>) is called '''abelian'''. If any two elements do not commute, the group is called '''nonabelian'''.

It is clear that Example 1 is an abelian group consisting of only two
elements <math>+1\,\!</math> and <math>-1\,\!</math>. However, Example 2 is clearly a nonabelian
group as can be seen from the multiplication table. For example
<math>\sigma_2R_2 = \sigma_1 \,\!</math>, but <math>R_2\sigma_2 =\sigma_3 \neq \sigma_1 \,\!</math>.

====Definition 4: Cyclic Group====
A '''cyclic group''' is a group in which every element of the group can be obtained from one element and all its distinct powers. The particular element is called the '''generating element'''.

Example 4 provides examples of cyclic groups.

====Definition 5: Subgroup====

A '''subgroup''' <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math> is a subset of the group elements that satisfies all
the properties in the definition of a group under the inherited multiplication rule.

====Example 3====

Consider the set <math>\{0,1,2,3, \cdots, N-1\}\,\!</math> and identify <math> N\,\!</math> and <math>0\,\!</math>. This is written as <math>0 \equiv N\,\!</math>. The operation on this set will be addition. This is the group of integers modulo <math>N\,\!</math> and is
denoted <math>\mathbb{Z}_N\,\!</math>. To be concrete, let us consider the group <math>\mathbb{Z}_3\,\!</math>, consisting of
<math>\{0,1,2;+\}\,\!</math>. (When the operation could be ambiguous, it is often useful to specify it explicitly along with the members of the set.) Let us check that this is a group. First, addition is certainly associative. Second, the identity is zero since
<math>a+0 =a\,\!</math> for any integer <math>a\,\!</math>. Third, <math>1+2=3 = 0\,\!</math> mod <math>3\,\!</math>.
In other words, since <math>3\,\!</math> and <math>0\,\!</math> are equivalent, the sum of one and
two is zero which is in the set. The order of the group is 3 (hence the subscript).

====Example 1 Revisited====

Recall [[#Example 1|Example 1]] is a group with <math>\{+1,-1\}\,\!</math> using multiplication.
This is the simplest nontrivial
''cyclic group'', since it is a cyclic group of order two.
All elements of this group are obtained from powers
of <math>-1\,\!</math>, namely <math>-1\,\!</math> and <math>(-1)^2 =1\,\!</math>. Notice that the generating
element is special; one cannot just take any element of the group to
be a generating element.

====Example 4====

We can represent the cyclic group of order <math>N\,\!</math>
in several ways. One we have seen is <math>\mathbb{Z}_N\,\!</math> with the operation of addition. Another is the set of elements
<math>\{e^{2\pi i n/(N-1)}\}\,\!</math>, <math>n = 0, 1, 2, 3, ..., N-1\,\!</math> with the operation of multiplication. Since this group can be
seen as the consisting of the element <math>e^{2\pi i/(N-1)}\,\!</math> and all its
powers, then this is a cyclic group with generating element
<math>e^{2\pi i/(N-1)}\,\!</math>.

====Example 5====

Include modular arithmetic under multiplication as a group.

===Comparing Groups: Homomorphisms and Isomorphisms===

Let us consider two groups <math>\mathcal{G}_1\,\!</math> and <math>\mathcal{G}_2\,\!</math> with product rules symbolized by <math>\cdot\,\!</math> and <math>\circ\,\!</math> respectively. Let the elements of <math>\mathcal{G}_1\,\!</math> be denoted <math>a_1,a_2, ...\,\!</math> and the elements of <math>\mathcal{G}_2\,\!</math> be denoted <math>b_1,b_2, ...\,\!</math> When comparing two groups to see how similar they are, the relationship among the
elements under the product rule is all-important. Therefore, if a map from one set of elements to another is given by <math>f:\mathcal{G}_1\rightarrow\mathcal{G}_2\,\!</math>, meaning <math>f(a_1) \in\mathcal{G}_2\,\!</math>, then the two groups have the same (algebraic) structure if, for all
<math>a_i,a_j,a_k \in \mathcal{G}_1\,\!</math>,
{{Equation|<math>
a_i\cdot a_j = a_k \;\; \Rightarrow \;\; f(a_i)\circ f(a_j) = f(a_k).
\,\!</math>|D.3}}
(Notice that this can be true even if the map takes all of the elements <math>a_i\,\!</math> to the identity.)

====Definition 6: Homomorphism====

If the condition [[#eqD.3|Eq.(D.3)]] is satisfied, the map is called a '''homomorpic map''' or a '''homomorphism'''. A homomorphism <math>f\,\!</math> satisfies the important property that
{{Equation|<math>
f(A\circ B) = f(A) \cdot f(B).
\,\!</math>|D.4}}
The composition <math>\circ\,\!</math> can, in general, be different from <math>\cdot\,\!</math>, but here both will be matrix multiplication unless otherwise stated.

====Definition 7: Isomorphism====

If a homomorphism is one-to-one (each <math>a_i\,\!</math> is mapped to one and only one <math>b_j\,\!</math>) and onto (each element in <math>\mathcal{G}_2\,\!</math> has an element of <math>\mathcal{G}_1\,\!</math> mapped to it), then the map is called an '''isomorphic map''' or an
'''isomorphism'''.

These definitions are used repeatedly in the representation theory of groups discussed below.

===Discussion===

With only these few definitions it is possible to discuss many important properties of groups and some of the reasons why they are so
important to physics. Let us first discuss some of the important properties of the group multiplication table.

====Group Multiplication Table====

The group multiplication table specifies the structure of the group and thus identifies a group. One example of this is when the
group is abelian. For all abelian groups the table is symmetric about the diagonal. (This follows from the fact that <math>ab=ba\,\!</math> for abelian
groups.) Another example is the presence of subgroups. This will be illustrated in this section.

====Subgroups: Return to Example 2====

In [[#Example 2|Example 2]], [[#Table D.1|Table D.1]] immediately shows that the elements
<math>\mathbb{I}, R_1,\,\!</math> and <math>R_2\,\!</math> form a subgroup since they are closed
under multiplication. Another somewhat less obvious subgroup
consists of the elements <math>\mathbb{I}\,\!</math> and <math>\sigma_1 \,\!</math>. This is a convenient
method for identifying subgroups, but is clearly limited to groups
with a relatively small order.

====The Rearrangement Theorem====

Notice that each group element appears in each row and each column of [[#Table D.1|Table D.1]] once and only once. This is no coincidence, but
is a general property of the multiplication table for groups. This implies that each row and column contains each and every group element
(due to the presence of the identity) so that each row and column is a simple rearrangement of the set of elements. For this reason, this is
sometimes called the rearrangement theorem and follows directly from the uniqueness of the elements in the set. (If there were two
elements in a row that were the same, then <math>ac=ab\,\!</math> for some <math>a,b,c\,\!</math>.
But then <math>a^{-1}ac = a^{-1}ab \Rightarrow c=b\,\!</math>, which cannot happen if
all elements are distinct.)

===A Little Representation Theory===

A group is specified by a set of elements, its product rule, and the
relations among the elements of the group under the product rule.
For finite order groups the group multiplication table is how one
identifies a group or shows that two groups are homomorphic
(explicitly or not).

====Definition 8: Representation====

A '''matrix representation''' of an abstract group is
any set of elements which is homomorphic to the set of elements in the abstract group.

More generally, if there is a homomorphic map from the set of abstract group elements onto a set of operators which, with their own combination rule (multiplication rule), satisfies the group axioms, then the operators form a representation of the group. (This includes preserving products as described in [[#Definition 6: Homomorphism|Section 3.1]].)

For our purposes, it is very important to note that a
set of group elements can always be represented by a set of matrices
so that we may restrict our attention to matrix representations.
Then, along with ordinary
matrix multiplication for the product rule, this provides a way to represent
any group. This is true for groups which have a finite order as well
as infinite order (discussed later).

Note that a representation is a ''homomorphism'' which can be a many-to-one map. If it is an isomorphism, the representation is said to be '''faithful'''. If, however, all matrices are the identity matrix, then all group elements are mapped to the identity, the multiplication relations (in the group multiplication table) are preserved and the representation is sometimes called the ''trivial representation''. This is always a valid, but not very informative, and certainly not faithful, representation of any group.

As will be shown in this first example, there are different sets of matrices which can represent the same group. This example will provide motivation for what follows.

====Example 6====

Let us consider an example of the representation of the group from
[[#Example 2|Example 2]]. This is a group of operations which will
take any permutation of the vertices to any other permutation. This
is also the set of permutations of three objects. This group is often
denoted <math>S_3\,\!</math>. The set of matrices representing the
rotations, reflection, and rotations combined with reflection provides
one way of representing this group. Another way to represent this
group is to use <math>3\times 3\,\!</math> matrices rather than the
<math>2\times 2\,\!</math> matrices given in the example. Let us
consider the following set of matrices.
{{Equation|<math>\begin{align}
\mathbb{I}_3 = \left(\begin{array}{ccc} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right), \;\;\;
&
P_2 = \left(\begin{array}{ccc} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}\right), \\
P_4 = \left(\begin{array}{ccc} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_1 = \left(\begin{array}{ccc} 1&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}\right),\\
P_3 = \left(\begin{array}{ccc} 0&0&1 \\ 0&1&0 \\ 1&0&0 \end{array}\right), \;\;\;
&
P_5 = \left(\begin{array}{ccc} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}\right).
\end{align}\,\!</math>|D.5}}
Clearly when these matrices act on a column vector labelling the
vertices,
{{Equation|<math>
\left(\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right).
\,\!</math>|D.6}}
the result is one of the permutations of three objects. These
orientations correspond to the same action as the <math>2\times 2\,\!</math> matrices
given in [[#Example 2|Example 2]] above. Therefore, these two sets of matrices
represent the ''same'' group, <math>S_3\,\!</math>. These representations are clearly
different. In fact the dimension of the matrices representing the
group is different for the two different representations. There are
other representations that can be immediately constructed. Consider
a set of matrices like the following:
{{Equation|<math>\begin{align}
\mathbb{I}_5 = \left(\begin{array}{cc} \mathbb{I}_3&0 \\ 0&\mathbb{I}_2 \end{array}\right), \;\;\;
&
g_2 = \left(\begin{array}{cc} P_2&0 \\ 0&R_1 \end{array}\right), \\
g_4 = \left(\begin{array}{cc} P_4&0 \\ 0&R_2 \end{array}\right), \; \;\;
&
g_1 = \left(\begin{array}{cc} P_1&0 \\ 0& \sigma_1 \end{array}\right), \\
g_3 = \left(\begin{array}{cc} P_3&0 \\ 0&\sigma_2 \end{array}\right), \;\;\;
&
g_5 = \left(\begin{array}{cc} P_5&0 \\ 0& \sigma_3 \end{array}\right).
\end{align}\,\!</math>|D.7}}
This set of matrices is said to be block-diagonal since it only has
non-zero elements in blocks along the diagonal. The <math>0\,\!</math> represents a
block of zeroes which is either <math>3\times 2\,\!</math> (upper right) or <math>2\times
3\,\!</math> (lower left). This set of matrices clearly satisfies the same multiplication relations as the sets given above
(<math>\{\mathbb{I}_3,P_1,P_2,P_3,P_4,P_5\}\,\!</math> and <math>\{\mathbb{I}_2, R_1, R_2, \sigma_1, \sigma_2, \sigma_3\}\,\!</math>) since the matrices multiply in
blocks. The elements of the group have the same multiplication table
and thus are isomorphic. Therefore this is another representation of
the group <math>S_3\,\!</math> which is different from either of the
two representations in the subblocks along the diagonal since it is a
combination of the two.

====Definition 9: Similarity Transformation====

Let <math>S\,\!</math> be an invertible matrix, and <math>M\,\!</math> be any matrix. In these notes, by '''similarity transformation''' we mean a transformation of the matrix <math>M\,\!</math> to <math>M^\prime\,\!</math>
{{Equation|<math>M^\prime = SMS^{-1}.\,\!</math>|D.8}}
We say the matrices <math>M\,\!</math> and <math>M^\prime\,\!</math> are similar matrices.

The importance of similarity transformations for representation theory is that they leave matrix equations unchanged. Suppose <math>A=BC \,\!</math>. Then defining <math>A^\prime = SAS^{-1}.\,\!</math>, <math>B^\prime = SBS^{-1}.\,\!</math>, and <math>C^\prime = SCS^{-1}.\,\!</math>,
<center>
<math>A=BC \; \Rightarrow A^\prime=B^\prime C^\prime\,\!</math>.
</center>

For more discussion about similarity transformations, see [[Appendix C - Vectors and Linear Algebra|Appendix C]], especially [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.5]], [[Appendix C - Vectors and Linear Algebra#The Trace|Section 3.6]], and [[Appendix C - Vectors and Linear Algebra#The Trace|Section 5.1]].

====Example 6 Continued====

However, Example 6 is a non-trivial problem even though the problem appears
trivial. The way to show that it is non-trivial is to
perform a similarity transformation, <math>g
\rightarrow S g S^{-1}\,\!</math> on all elements <math>g\,\!</math>
of the group. Since <math>S \,\!</math> is any invertible matrix,
it could mix all rows and columns. This would make it very difficult to identify the block-diagonal form or even know that it exists unless some other tools are used.

Furthermore, given a set of matrices which are known to form a representation of the group, it is non-trivial to find the similarity transformation which will simultaneously block-diagonalize all of these matrices to enable the identification of irreducible blocks.

====Equivalent Representations====

Two representations, <math>D^{(1)}\,\!</math> and <math>D^{(2)} \,\!</math> are '''equivalent''' if and only if there is an invertible matrix <math>S \,\!</math>, such that <math>D^{(1)} = SD^{(2)}S^{-1} \,\!</math>, otherwise they are inequivalent.

We will only consider matrix representations. In this case, the matrices will act on a vector space <math>\mathcal{V}\,\!</math>, called the '''representation space.'''

===Miscellaneous Definitions===

====Definition 10: Stabilizer====

The '''stabilizer''' of an element <math>\mathcal{m}\,\!</math> of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leave the element <math>m\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}
\,\!</math>|D.9}}
The stabilizer of <math>m\,\!</math> is also called the '''isotropy group''' of <math>m\,\!</math>, the '''isotropy subgroup''' of <math>m\,\!</math>, the '''stationary subgroup''' of <math>m\,\!</math>, or in physics, sometimes the '''little group''' of <math>m\,\!</math>.

====Definition 11: Centralizer====

The '''centralizer''' subgroup of a group consists of elements of the group that commute with all elements of a certain set.

====Definition 12: Pauli Group====
The '''Pauli Group''' on <math>n\,\!</math> qubits, denoted <math>\mathcal{P}_n\,\!</math>, is the set of <math>n\,\!</math> tensor products of the Pauli matrices <math>\mathbb{I}, X, Y, Z\,\!</math> along with coefficients <math>\pm 1,\pm i\,\!</math>. This is an example of a group. It is defined here due to its importance for quantum error correcting codes and the factors <math>\pm 1,\pm i\,\!</math> are required for the closure property in the definition of a group.

====Properties of the Pauli Group====

Let us consider the Pauli group for 2 qubits, but omit the tensor product symbols. The following are elements
{{Equation|<math>
\mathbb{I}\mathbb{I},\mathbb{I}X,\mathbb{I}Y,\mathbb{I}Z,X\mathbb{I},XX,XY,XZ,Y\mathbb{I},YX,YY,YZ,Z\mathbb{I},ZX,ZY,ZZ,
\,\!</math>|D.10}}
as are all of these elements multiplied by <math>-1,\,\!</math> and all of these elements multiplied by <math>i,\,\!</math> as well as all of these elements multiplied by <math>-i.\,\!</math> Thus there are <math>4^3\,\!</math> total elements of the group for two qubits. In general there are <math>4\cdot 4^n\,\!</math> elements for the Pauli group for <math>n\,\!</math> qubits.

One of the nice and interesting properties of the Pauli group is that every pair, say <math>A,B\,\!</math> of elements of the Pauli group either commutes <math>[A,B]= AB-BA =0\,\!</math> or anti-commutes <math>\{A,B\} = AB+BA =0\,\!</math>. This turns out the be very useful.

Another notation for [[#eqD.10|Equation (D.10)]] is
{{Equation|<math>
\mathbb{I},X_2,Y_2,Z_2,X_1,X_1X_2,X_1Y_2,X_1Z_2,Y_1,Y_1X_2,Y_1Y_2,Y_1Z_2,Z_1,Z_1X_2,Z_1Y_2,Z_1Z_2,
\,\!</math>|D.11}}
Clearly this index notation has an advantage for large products. It also has the advantage that it enables us to immediately see the weight of an operator.

====Definition 13: Weight of an Operator====

The '''weight of an operator''' is the number of non-identity elements in the tensor product.

This definition is most often used in the context of the Pauli Group. It's importance is seen in quantum error correcting codes.

====Definition 14: Generators of a Group====

Let us consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements which, through multiplication will give all of the (sub)group elements. The elements in this subset are called '''generators''' of the group.

Note that the set of generators is not unique.

The generators are a very convenient set to use because it is a much smaller set than the whole group and many properties of the group can discovered using only the generators. For example, if every generator of a subgroup acts on an object and leaves it invariant, then every element of the group will also leave it invariant since they are all given by products of the generators. Thus one only needs to check whether or not the generators will leave an object invariant.

One example is the stabilizer subgroup where a set of generators stabilizes, or leaves invariant, the code words of the stabilizer code.

====Definition 15: Normalizer====

The '''normalizer''' of a set <math>\mathcal{M}\,\!</math> is the subgroup <math>\mathcal{S}\,\!</math> of a group <math>\mathcal{G}\,\!</math>, that leaves the set <math>\mathcal{M}\,\!</math> fixed:
{{Equation|<math>
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.
\,\!</math>|D.12}}
Note the difference between the centralizer, with which this should not be confused. The centralizer leaves ''every element'' of the set fixed. The elements of the normalizer contain the elements of the centralizer as a special case, but they can move elements around within the set.

====Definition 16: Coset====

Let <math>\mathcal{S}\,\!</math> be a subgroup of a group <math>\mathcal{G}\,\!</math> and let <math>G\,\!</math> an element of <math>\mathcal{G}\,\!</math>. The left '''coset''' is a subset of the group
{{Equation|<math>
G\mathcal{S} = \{GS|S\in\mathcal{S} \}.
\,\!</math>|D.13}}

One can similarly define the right coset.

The importance of cosets is that the partition the group in a particular way. If there is another coset,
{{Equation|<math>
K\mathcal{S} = \{KS|S\in\mathcal{S} \},
\,\!</math>|D.14}}
then either <math>G\mathcal{S}=K\mathcal{S}\,\!</math> or they are disjoint sets, having no element in common. (This is because <math>\mathcal{S}\,\!</math> is a subgroup. So you could multiply by an element to show they are the same set.)

===Infinite Order Groups: Lie Groups===

All of the examples presented so far have been groups with finite order. Groups which have infinite order, can be described with one or more parameters, and are differentiable with respect to those parameters, are called ''Lie groups''.

====Definition 17: Lie Group====

A '''Lie group''' is a group which is also a differentiable manifold. (See for example [[Bibliography#Cecile:book|Analysis, Manifolds, and Physics]]).

In this section, several examples of Lie groups are given. In physics these groups correspond to a continuous set of symmetries, whereas the groups of finite order correspond to a discrete set of symmetries.

====Example 7====

The Lie group most often used as the introductory example is the group consisting of the set <math>e^{i\theta}\,\!</math>, for all
<math>\theta\,\!</math>. This group has an infinite number of elements (an infinite order), and has one parameter, <math>\theta\,\!</math>. The group is also a differentiable manifold, a circle. Notice this group is also isomorphic to the set of matrices
{{Equation|<math>
\left(\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right).
\,\!</math>|D.15}}
If this matrix were to act on a unit vector in the x-y plane, it would rotate that vector around in a circle and after <math>2\pi\,\!</math>, the tip of the vector would sweep out a circle of unit radius.

====Example 8====

Another example of a Lie group, and one of the most important for quantum information, is the set of complex, <math>2\times 2\,\!</math> matrices,
which satisify,
{{Equation|<math>
U^\dagger U = \mathbb{I} = U U^\dagger.
\,\!</math>|D.16}}
This group is called <math>U(2)\,\!</math>. This is called the set of ''unitary''
<math>2\times 2\,\!</math> matrices. (Hence the <math>U\,\!</math>.) Notice that the determinant
of this set is <math>e^{i\alpha}\,\!</math>, where <math>\alpha\,\!</math> is a real number, since
{{Equation|<math>
1 = \det(\mathbb{I}) = \det(U U^\dagger) = \det(U)\det(U^\dagger)
= \det(U)(\det(U))^*.
\,\!</math>|D.17}}

There is a subgroup of this group which is often considered. It is the subgroup with determinant one. This group is denoted <math>SU(2)\,\!</math>,
and is called the ''special unitary group''. The term unitary refers to the fact that <math>U^\dagger U = I = UU^\dagger\,\!</math> and the "S", for special indicates that it has determinant one.

====Example 9====

One can immediately generalize the unitary and special unitary groups
to <math>N\times N\,\!</math> matrices. These are denoted <math>U(N)\,\!</math> and <math>SU(N)\,\!</math>
respectively. In quantum computing a set of unitary groups which is important is the set with <math>U(2^n)\,\!</math> where <math>n\,\!</math> is the number of qubits. This is the set of all possible unitary transformations on a set of qubits.

====Example 10====

The complex General Linear group is the set of invertible <math>N\times N\,\!</math> matrices with complex numbers as entries. It is denoted <math>GL(N,\mathbb{C})\,\!</math>.

===More Representation Theory===

In Physics we are most often concerned with linear representations of groups which use linear operators to represent group elements. These linear operators are represented by matrices in physics. In this Appendix, the focus is entirely on these types of representations although this is not always stated explicitly. Although these comments have been made above for finite groups, they are worth reiterating due to their importance and because they also apply to infinite order groups, such as Lie groups. Furthermore, definitions introduced for finite order groups are also applicable to Lie groups.

Thus the previous discussion of representation theory applies to the representation of Lie groups. A representation of a group can be "reduced" to block-diagonal form. When these blocks cannot be further reduced, the blocks are called "irreducible". These irreducible blocks which make up "irreducible representations" and our study of representation theory is our concern with irreducible blocks and how to find them.

Clearly a set of matrices which may be block-diagonalizable, but which have been acted upon by a highly non-trivial <math>S \,\!</math> acting on them may well represent a group <math>\mathcal{G}\,\!</math> for sets of matrices with many different dimensions and many different block-diagonal forms. Therefore finding irreducible blocks and the similarity transformation that simultaneously block-diagonalizes all matrices of a given representation is highly non-trivial.

Before discussing the representation of Lie groups, there is another definition that is quite helpful.

====The Lie Algebra of a Lie Group====

The Lie algebra of a Lie group is defined as the set of left-invariant vector fields on the manifold of the Lie group. For our purposes, the Lie algebra will be described by the basis elements of the tangent space to the origin of the group which is isomorphic to the set of left-invariant vector fields. To see how to relate the group and algebra and to see how this is useful, let us suppose there is a Lie algebra corresponding to a Lie group which has a set of basis elements <math>\{\lambda_i\}\,\!</math>. To describe the relation between the Lie group and Lie algebra, let <math>g\in\mathcal{G}\,\!</math> and let <math>\{a_i\}\,\!</math> be a set of parameters (which can be taken to be real). Then an element of the Lie algebra is given by <math> \sum_i a_i\lambda_i\,\!</math> and an element of the group can be written in terms of these parameters as
{{Equation|<math>
g=\exp\left(-i\sum_i a_i \lambda_i\right).
\,\!</math>|D.15}}
The tangent space to the origin is given by the derivative of <math>g \,\!</math> with respect to the parameters <math> a_i\,\!</math>. In this way, one sees that the group is an analytic manifold. There are several reasons why it is useful to consider the Lie algebra. One is that it is often easier to analyze than the Lie group and several important properties of the Lie group are able to be obtained from properties of the Lie algebra. For example subalgebras correspond to subgroups.

====Representation Theory for Lie Groups====

As with finite order groups, one of the primary objectives of this introduction to group theory is to enable one to find irreducible representations of a group from a given reducible one. At the least, the objective should be to understand what this means, how one would go about it in principle, and how it is used in quantum physics and quantum computing.

Lie groups, represented by a set of matrices consisting of differentiable parameters, may also be described by matrices which are reducible to block-diagonal form to blocks which cannot be reduced further. These irreducible blocks form irreducible representations of the group. One may suppose that irreducible representations of Lie groups are more difficult to understand than finite groups due to the fact that there are an infinite number of matrices in the set of group elements. This is certainly true, so one sometimes relies on the Lie algebra. Suppose a set of elements <math>\{\lambda_i\}\,\!</math> of a Lie algebra obeys a particular set of commutation relations, say
{{Equation|<math>
[\lambda_i,\lambda_j] = 2i\sum_kf_{ijk}\lambda_k,
\,\!</math>|D.16}}
where <math>f_{ijk}\,\!</math> is some set of constants (and the factor of two is a non-standard convention). Then any other set which obeys the same commutation relations is also a representation of the same Lie algebra. The representation of the algebra can then give a representation of the group through exponentiation although the representation may not be faithful.

Now let us suppose that there exists a similarity transformation <math>S\,\!</math> that will simultaneously block-diagonalize all elements of a group. Then, observing that
{{Equation|<math>
SgS^{-1}=S\exp\left(-i\sum_i a_i \lambda_i\right)S^{-1} = \exp\left(-i\sum_i a_i S\lambda_iS^{-1}\right),
\,\!</math>|D.17}}
it is clear that the same similarity transformation will block-diagonalize the elements of the algebra as well.

====Some Useful Relations Among Lie Algebra Elements====

A Lie algebra will obey the the commutation relations, [[#eqD.16|Equation (D.16)]]. However, since the emphasis here is the representation of groups in terms of matrices, there are several other useful relations that will be listed. These relations apply to all Lie algebra elements of <math>SU(d)\,\!</math>.

We have chosen the following convention for the normalization of
the algebra of Hermitian matrices which represent generators of <math>SU(d)\,\!</math>.
{{Equation|<math>
\text{Tr}(\lambda_i\lambda_j) = 2\delta_{ij}.
\,\!</math>|D.18}}

The commutation and anti-commutation relations of the matrices
representing the basis for the Lie algebra can be summarized
by the following equation:
{{Equation|<math>
\lambda_i \lambda_j = \frac{2}{d}\delta_{ij} + if _{ijk} \lambda_k
+ d_{ijk}\lambda_k,
\,\!</math>|D.19}}
where here, and throughout this section, a sum over repeated
indices is understood.

As with any Lie algebra we have the Jacobi identity:
{{Equation|<math>
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0.
\,\!</math>|D.20}}
which may also be written as
{{Equation|<math>
[[\lambda_i,\lambda_j],\lambda_k]+ [[\lambda_j,\lambda_k],\lambda_i] + [[\lambda_k,\lambda_i],\lambda_j] =0.
\,\!</math>|D.21}}
There is also a Jacobi-like identity,
{{Equation|<math>
f_{ilm}d_{jkl} + f_{jlm}d_{kil} + f_{klm}d_{ijl} =0,
\,\!</math>|D.22}}
which was given by Macfarlane, et al. \cite{Macfarlane}.

The following identities, also provided in cite{Macfarlane},
{{Equation|<math>
\begin{align}
d_{iik} &= 0, \\
d_{ijk}f_{ljk} &= 0, \\
f_{ijk}f_{ljk} &= d\delta_{il}, \\
d_{ijk}d_{ljk} &= \frac{d^2 - 4}{d}\delta_{il},
\end{align}
\,\!</math>|D.23}}
and
{{Equation|<math>
f_{ijm}f_{klm} = \frac{2}{d}(\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk})
+ (d_{ikm}d_{jlm} - d_{jkm}d_{ilm})
\,\!</math>|D.24}}
and finally
{{Equation|<math>
\begin{align}
f_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)f_{ijk},\\
d_{piq}f_{qjr}f_{rkp} &= -\left(\frac{d}{2}\right)d_{ijk},\\
d_{piq}d_{qjr}f_{rkp} &= \left(\frac{d^2 - 4}{2d}\right)f_{ijk},\\
d_{piq}d_{qjr}d_{rkp} &= \left(\frac{d^2 - 12}{2d}\right)d_{ijk}.
\end{align}
\,\!</math>|D.25}}
The proofs of these are fairly straight-forward and are omitted.

====Tensor Products of Representations====

When one takes the tensor product of two representations, another representation results. In general, this representation is reducible.

To see this, let <math> g_1,g_2,g_3,g_4 \in \mathcal{G}\,\!</math>. A tensor product of two group elements is <math> g_1\otimes g_1 \in \mathcal{G}\otimes \mathcal{G} \,\!</math>. Certainly, when <math> g_1\cdot g_2 = g_3, \,\!</math> then <math> g_1\otimes g_1\cdot g_2 \otimes g_3 = g_3 \otimes g_3\,\!</math>. (See [[Appendix C - Vectors and Linear Algebra#Tensor Products|Section C.7]].) Therefore, the tensor product of two representations is another representation. However, even if this is an irreducible representation, one would suspect that the tensor product is a reducible representation, and this turns out to be true. The task is to find the irreducible components.

One very important example of this is used for the addition of angular momenta. Before revisiting the more general case, this important example is discussed.

====Addition of Angular Momenta====

In the theory of angular momenta, quantum states are labelled according to their total angular momentum eigenstates and the z component of their angular momentum. Let the total angular momentum square be given by the operator
{{Equation|<math>
\vec{J} = (J_x,J_y,J_z).
\,\!</math>|D.26}}
These operators satisfy the commutation relations
{{Equation|<math>
[J_i,J_j] = i\epsilon_{ijk}J_k,
\,\!</math>|D.27}}
where <math>i,j,k = 1,2,\text{ or }3, \,\!</math> and the epsilon tensor is defined in [[Appendix C - Vectors and Linear Algebra#eqC.9|Equation C.9]]. A state <math>\left| j, m\right\rangle\,\!</math> satisfies
{{Equation|<math>
J^2\left| j, m\right\rangle = j(j+1)\hbar^2\left| j, m\right\rangle, \;\; \text{and} \;\; J_z\left| j, m\right\rangle = m\hbar\left| j, m\right\rangle,
\,\!</math>|D.28}}
where
{{Equation|<math>
J^2 = \vec{J} \cdot \vec{J} = J_x^2 + J_y^2 + J_z^2.
\,\!</math>|D.29}}
The common problem is as follows. Given two states <math>\left| j_1, m_1\right\rangle\,\!</math> and <math>\left| j_2, m_2\right\rangle\,\!</math>, find the total angular momentum of the two states combined. The objective, is to find a new basis, <math>\left| j, m, j_1, j_2\right\rangle\,\!</math>, which is expressed in terms of the old basis. In other words, a set of numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> such that
{{Equation|<math>
\left| j, m, j_1, j_2\right\rangle = \sum_{m_1,m_2} C(j_1,j_2,j,m|m_1,m_2,m) \left| j_1, m_1\right\rangle \left| j_2, m_2\right\rangle.
\,\!</math>|D.30}}
The numbers <math>C(j_1,j_2,j,m|m_1,m_2,m)\,\!</math> are called Clebsch-Gordan coefficients, or Wigner-Clebsch-Gordan coefficients, and not only put the tensor product of the vectors into this special form, they also block-diagonalize the tensor products of the operators. The most common example of this is the addition of angular momentum of two spin-1/2 particles. The result is a triplet (spin-1 representation) and a singlet (spin-0 representation).

<math> \,\!</math>

====Concluding Remarks====

'''To summarize''', matrix representations of a group are sets matrices which represent the group in the sense that they follow the same multiplication law as the original group elements. The representation may be reducible, meaning the set of matrices may be all block-diagonalized by a single similarity transformation such that each individual block will represent a group element in its own representation. If a representation (the set of matrices) cannot be transformed into by a single similarity transformation such that each matrix is comprised of a set of smaller blocks, then the representation is called irreducible. If there is a isomorphism from the set of matrices to the original group, then the representation is faithful.