https://www2.physics.siu.edu/qunet/wiki/api.php?action=feedcontributions&feedformat=atom&user=RceballosQunet - User contributions [en]2026-04-29T19:12:59ZUser contributionsMediaWiki 1.31.7https://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=905Index2010-07-07T22:34:45Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:Dirac notation [[Appendix C - Vectors and Linear Algebra#Introduction|'''C.2.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean (see Average)<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]]<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transformation [[Chapter 1#Bits and qubits: An Introduction|1.3]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Circuit Diagrams for Qubit Gates|2.3.1]], [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]], [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 8 - Noise in Quantum Systems#Modelling Open System Evolution|8.3]], [[Chapter 8 - Noise in Quantum Systems#Fixed-Basis Operations|8.3.2]], [[Chapter 8 - Noise in Quantum Systems#Unitary Freedom|8.4.1]], [[Chapter 8 - Noise in Quantum Systems#Physical Interpretation of the Unitary Freedom|8.4.2]], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']], [[Appendix D - Group Theory#Introduction|'''D.1''']]<br />
::active [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
::passive [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Appendix_B_-_Complex_Numbers&diff=900Talk:Appendix B - Complex Numbers2010-05-03T03:17:53Z<p>Rceballos: </p>
<hr />
<div>I think when you state that y and z are the imaginary parts of a complex number, you really meant "y and d" are the imaginary parts. I also think that the note referring to the product of complex conjugates leaves a little room for confusion when you start back up afterward with, "we call this modulus squared." I just ended up moving the "note" down a sentence. I think this way is more clear. Russ.</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_B_-_Complex_Numbers&diff=899Appendix B - Complex Numbers2010-05-03T03:16:39Z<p>Rceballos: </p>
<hr />
<div>Complex numbers arise naturally from an attempt to solve the equation <br />
<center><math><br />
a^2 + 1 = 0.<br />
\,\!</math></center><br />
It's easy enough to write such an equation down, but how would you<br />
solve it? The answer is<br />
<center><math><br />
a = \sqrt{-1} =i.<br />
\,\!</math></center><br />
We let the symbol <math>i\,\!</math> represent <math>\sqrt{-1}\,\!</math>, so that <math>i^2=-1\,\!</math>. Then<br />
any number of the form <br />
<center><math><br />
z = x + iy, <br />
\,\!</math></center><br />
where <math>x\,\!</math> and <math>y\,\!</math> are real, is called a ''complex'' number.<br />
<!-- \index{complex number} --> <br />
Let's take some other complex number to be <math>\eta = c+ id\,\!</math>, where <math>c\,\!</math> and <math>d\,\!</math> are real. Then the two complex numbers are equal, <br />
<center><math><br />
\eta = z <br />
\,\!</math></center><br />
which is to say<br />
<center><math><br />
x +iy = c+id, <br />
\,\!</math></center><br />
if and only if<br />
<center><math><br />
x = c, \;\;\mbox{ and } \;\; y = d.<br />
\,\!</math></center><br />
<br />
We refer to <math>x\,\!</math> as the ''real part'' of the complex number <math>z\,\!</math> and<br />
<math>y\,\!</math> as the ''complex part''. Sometimes these are written as Re(<math>z\,\!</math>)<br />
and Im(<math>z\,\!</math>), respectively. <br />
<br />
We may restate the equivalence condition as <math>z=\eta\,\!</math> if and only if<br />
the real part of <math>z\,\!</math> is equal to the real part of<br />
<math>\eta\,\!</math> ''and'' the imaginary part of <math>z\,\!</math> is equal to the imaginary part of <math>\eta\,\!</math>. <br />
<br />
Complex numbers are multiplied like any other binomial expression:<br />
<center><math><br />
z\eta = (x+iy)(c+id) = xc - yd +i(yc + xd), <br />
\,\!</math></center><br />
where we have used <math>i^2 = -1\,\!</math>. <br />
<br />
The ''complex conjugate'' <!-- \index{complex conjugate} -->of the complex<br />
number <math>z\,\!</math> is denoted <math>z^*\,\!</math> and is given by<br />
<center><math><br />
z^* = x-iy. <br />
\,\!</math></center><br />
One reason for defining this is that a number times its own complex<br />
conjugate is real,<br />
<center><math><br />
zz^* = (x+iy)(x-iy) = x^2 + y^2 +i(xy - yx) = x^2 +y^2. <br />
\,\!</math></center><br />
Note that the complex conjugate of the complex conjugate is the<br />
original complex number and <br />
<center><math><br />
z^*z = (x+iy)(x-iy) = x^2 + y^2 +i(xy - yx) = x^2 +y^2 = zz^*. <br />
\,\!</math></center><br />
<br />
We also call this the ''modulus squared'' <!-- \index{modulus squared}--> so<br />
that the ''modulus'' is<br />
<center><math><br />
|z| = \sqrt{(z^*z)} = \sqrt{x^2 + y^2}.<br />
\,\!</math></center><br />
Note that the complex conjugate of a product is the product of the complex<br />
conjugates: <br />
<center><math><br />
(z\eta)^* = z^* \eta^*.<br />
\,\!</math></center><br />
<br />
<br />
<br />
It is often useful to look at a graph for a complex number. The graph<br />
consists of an x-axis for the real part, and a y-axis for the<br />
complex part. This is shown in [[#Figure B.1|Fig. B.1]]. In this<br />
figure, it is easily seen that we can think of <math>z\,\!</math> as a<br />
two-dimensional vector and that the magnitude (length) of the vector<br />
is the modulus of the complex number, <math>|z| = \sqrt{x^2 + y^2}\,\!</math>.<br />
<div id="Figure B.1"></div><br />
<center><br />
[[File:complexgraph1.jpeg]]<br />
Figure B.1: A complex number in Cartesian coordinates.</center><br />
<br /><br />
<br />
<!-- \begin{figure}<br />
\begin{center}<br />
\includegraphics[scale=0.5]{/home/mbyrd/tex/books/qcomp/figures/complexgraph1.eps}<br />
\caption{\label{fig:compg1} A Cartesian coordinate representation of<br />
a complex number <math>z\,\!</math>.}<br />
\end{center}<br />
\end{figure} --><br />
<br />
<br />
Another useful way to represent this is with polar coordinates. We<br />
can do this by writing <br />
<center><math><br />
z = |z|(\cos\theta +i \sin\theta).<br />
\,\!</math></center><br />
It turns out that <br />
<center><math><br />
e^{i\theta} = \cos\theta + i \sin\theta,<br />
\,\!</math></center><br />
so we could also write<br />
<center><math><br />
z = |z|e^{i\theta}.<br />
\,\!</math></center><br />
It is often the case that people will write this as <br />
<center><math><br />
z = re^{i\theta},<br />
\,\!</math></center><br />
where <math>r = \sqrt{x^2+y^2}\,\!</math> as is usual for polar coordinates. Then,<br />
everything is just like polar coordinates, with the exception of the<br />
inclusion of the factor <math>i\,\!</math>. (See [[#Figure B.2|Fig. B.2]].) <br />
<div id="Figure B.2"></div><br />
<center><br />
[[File:complexgraph2.jpeg]]<br />
Figure B.2: A polar coordinate representation of a complex number.</center><br />
<br /></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_F_-_NOTES_and_CREDITS&diff=898Appendix F - NOTES and CREDITS2010-05-03T03:12:14Z<p>Rceballos: </p>
<hr />
<div>By now there seem to be quite a few books on quantum computing. However, there are only a<br />
few which were recommended for the course. (These are in no particular order.)<br />
<br />
*Visit http://www.qubit.org and click on “What is QIP?” There are several introductory articles there about quantum mechanics and quantum computing. There are other interesting articles as well.<br />
*N. David Mermin’s book [11].<br />
*:Paraphrasing from the book: This book was written as an introduction to quantum computation which does not assume any background in physics. It evolved from his course on quantum computing for undergraduate and graduate students at Cornell and is for students from computer science, mathematics, engineering, and physics.<br />
*:I have heard him speak and read another book of his. His explanations are great.<br />
*Shankar’s book: Principles of Quantum Mechanics<br />
*:The book contains topics which are more advanced, but starts fairly simply and the primary reason I list it here is that the first chapter has a lot of the mathematics that we will be using (and probably more) written at a fairly basic level, although it is somewhat abstract.<br />
*Michael Nielsen and Isaac Chuang’s book [13].<br />
*:This book has a little bit about many different subjects. Its easy to read (for the most part) and contains very important and fairly basic information. This has been THE textbook and reference book since about 2001.<br />
*John Preskill’s Course notes [9].<br />
*:Amazing! These notes were essentially THE textbook treatment before the book of Neilsen and Chuang book and are still very good for a variety of introductory topics as well as a good reference for a variety of topics. Some of them quite advanced. It’s available on the web!<br />
*Frank Gaitan’s book [5].<br />
*:To my knowledge this is the first book on quantum error correction. Approximately the second half of the course will cover quantum error prevention methods.<br />
<br />
<br />
<br />
By the time these notes, and the course, are finished, several people will have contributed.<br />
These are listed here:<br />
*Mark S. Byrd<br />
*C. Allen Bishop <br />
*Nayeli Zuniga-Hansen<br />
*Seyoum Tsige <br />
*Max Herlache <br />
*Philip Feinsilver<br />
*Jalal Alowibdi <br />
*Sarah Harvey<br />
*Kevin Reuter<br />
*Russell Ceballos</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=897Appendix C - Vectors and Linear Algebra2010-05-03T02:11:40Z<p>Rceballos: /* Hermitian Conjugate */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by a hat (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors. <!-- \index{basis vectors!real} --> <br />
This is because any vector<br />
(in real three-dimensional space) can be written in terms of these unit/basis vectors. In<br />
some sense they are the basic components of any vector. Other basis<br />
vectors could be use, however. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in higher dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will briefly discuss several of these aspects here,<br />
however, we will first provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose of a matrix is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, then<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector and in Dirac notation, the symbol <math>\left\langle\cdot \right\vert\!</math>, is called a ''bra''. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and <br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C.16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>. (The <math> \left\langle\phi\mid\psi\right\rangle \,\!</math><br />
is called a ''bracket'' which is the product of the ''bra'' and the ''ket''.) The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, <br />
often when one vector is transformed to another, the transformation can be viewed as a transformation of the components of the vector and is <br />
also represented by a matrix. Thus transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
In the case that we consider a matrix transformation of basis elements, we call it a passive transformation. (The transformation does nothing to the object, but only changes the basis in which the object is described.) An active transformation is one where the object itself is transformed. Often these two transformations, active and passive, are very simply related. However, the distinction can be very important. <br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
<br />
====Example 1====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
====Example 2====<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']] are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
====Example 3====<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right) <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Appendix_B_-_Complex_Numbers&diff=896Talk:Appendix B - Complex Numbers2010-05-03T01:55:23Z<p>Rceballos: </p>
<hr />
<div>I think when you state that y and z are the imaginary parts of a complex number, you really meant "y and d" are the imaginary parts. I also think that the note referring to the product of complex conjugates leaves a little room for confusion when you start back up afterward with, "we call this modulus squared." Perhaps I should include a zz* at the beginning of the sentence to pick up your train of thought again. Russ.</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Appendix_B_-_Complex_Numbers&diff=895Talk:Appendix B - Complex Numbers2010-05-03T01:49:44Z<p>Rceballos: Created page with 'I think when you state that y and z are the imaginary parts of a complex number, you really meant "y and d" are the imaginary parts. Russ.'</p>
<hr />
<div>I think when you state that y and z are the imaginary parts of a complex number, you really meant "y and d" are the imaginary parts. Russ.</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Appendix_A_-_Basic_Probability_Concepts&diff=894Talk:Appendix A - Basic Probability Concepts2010-05-03T01:45:22Z<p>Rceballos: Created page with 'I think that the only thing this section may need, is a little more explicit definition of J towards the beginning or middle of the section. But other than that, I think its as …'</p>
<hr />
<div>I think that the only thing this section may need, is a little more explicit definition of J towards the beginning or middle of the section. But other than that, I think its as concise and clear as it can be. Russ.</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_A_-_Basic_Probability_Concepts&diff=893Appendix A - Basic Probability Concepts2010-05-03T00:52:52Z<p>Rceballos: </p>
<hr />
<div>In this appendix definitions and some example calculations are<br />
presented which will aid in our discussions. This is not meant to be<br />
a comprehensive introduction to the topic. It is primarily meant to<br />
serve as a means for introducing notation and terminology for the<br />
course. This example is a variation of one given by David Griffiths<br />
in ''Intoduction to Quantum Mechanics'' ([[Bibliography#Griffiths:qmbook|David J. Giffiths’s book]]).<br />
<br />
''Example'': Suppose that in some room, there are four people. Their heights in meters are: <br />
#1 person is 1.5 meters tall<br />
#1 person is 1.6 meters tall<br />
#2 people are 1.8 meters tall<br />
We might write this as (<math>N\,\!</math> will stand for the number of people) <br />
<math>N(1.5) = 1\,\!</math>, <math>N(1.6)=1\,\!</math>, <math>N(1.8)=2\,\!</math> and the total number of people is <br />
<center><math><br />
N = \sum_{j=0}^\infty N(j),<br />
\,\!</math></center><br />
where <math>j\,\!</math> runs over all values and in this case we are rounding to the<br />
nearest tenth of a meter. Here <math>N\,\!</math> =4 of course. <br />
<br />
Now if I draw a name out of a hat which contains each persons name<br />
once, I will get the persons name which is 1.6 meters tall with<br />
probability <math>1/4\,\!</math>. (We assume that each person has a unique name and<br />
that it appears once and only once in the hat.) We write this as<br />
<center><math><br />
P(1.6) = 1/4,<br />
\,\!</math></center><br />
and we would generally write for any value<br />
<center><math><br />
P(j) = \frac{N(j)}{N}. <br />
\,\!</math></center><br />
Now since we are going to get someone's name when we draw, we must<br />
have <br />
<center><math><br />
\sum_j P(j) = 1,<br />
\,\!</math></center><br />
which is easy enough to check. <br />
<br />
There are several aspects of this probability distribution that we might like to know. Here are some which are particularly useful: <!-- \index{median}\index{mean} \index{average}--><br />
#The ''most probable'' values for the height is 1.8 meters.<br />
#The ''median'' is 1.7 meters (two people below, and two above)<br />
#The ''average'' (or ''mean'') is given by<br />
{{Equation|<math>\begin{align}\left\langle height\right\rangle &= \frac{1(1.5)+1(1.6)+2(1.8)}{4} \\ &= \frac{6.8}{4} = 1.7 \end{align}</math>|A.1}} <br />
Note that the mean and the median do not have to be the same. The<br />
median is the middle number in the list, if there is an odd number,<br />
otherwise it is the mean of the two in the middle. These two just<br />
happen to be the same here. <br />
The bracket <math>\left\langle\cdot\right\rangle\,\!</math> is fairly standard notation and it means<br />
generally, that we obtain the ''average value''<!-- \index{average}--> <br />
of a function by calculating <br />
<center><math>\left\langle f(j)\right\rangle = \sum_{j=0}^\infty f(j)P(j).\,\!</math></center><br />
For the average this is just <br />
<center><math><br />
\left\langle j\right\rangle = \sum_{j=0}^\infty jP(j)= \sum_{j=0}^\infty j\frac{N(j)}{N}. \,\!</math></center><br />
<br />
'''Note:''' The ''average value'' is called the ''expectation value'' <!-- \index{expectation value} --> in quantum mechanics. This can be<br />
misleading becase it is ''not'' the most probable and it is not <nowiki>''what to expect.''</nowiki> <br />
<br />
When one would like to discuss a properties of a particular probability distribution, describing it takes some effort. It is not enough to know the average, median, and most probable values. This leaves a lot of details of the probability distribution unknown to us if these are all we are given. What else would one like to know? Without describing it entirely, one may like to know more about the <nowiki>''shape''</nowiki> of the distribution. For example, how spread out is it?<br />
<br />
The most important measure of this is the ''variance'',<!-- \index{variance}--> which is the ''standard deviation'', <!-- \index{standard deviation} --> squared. The variance is defined as (in terms of our variable <math>j\,\!</math>) <br />
{{Equation|<math>\sigma^2 = \langle(\Delta j)^2\rangle, \,\!</math>|A.2}}<br />
where <math>\Delta j = j -\langle j \rangle\,\!</math>. This can also be written as <br />
{{Equation|<math>\sigma^2 = \langle j^2\rangle - \langle j \rangle^2.\,\!</math>|A.3}}</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=892Index2010-05-03T00:43:30Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:Dirac notation [[Appendix C - Vectors and Linear Algebra#Introduction|'''C.2.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]]<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transformation [[Chapter 1#Bits and qubits: An Introduction|1.3]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Circuit Diagrams for Qubit Gates|2.3.1]], [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]], [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 8 - Noise in Quantum Systems#Modelling Open System Evolution|8.3]], [[Chapter 8 - Noise in Quantum Systems#Fixed-Basis Operations|8.3.2]], [[Chapter 8 - Noise in Quantum Systems#Unitary Freedom|8.4.1]], [[Chapter 8 - Noise in Quantum Systems#Physical Interpretation of the Unitary Freedom|8.4.2]], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']], [[Appendix D - Group Theory#Introduction|'''D.1''']]<br />
::active [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
::passive [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=891Index2010-05-03T00:37:45Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:Dirac notation [[Appendix C - Vectors and Linear Algebra#Introduction|'''C.2.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transformation [[Chapter 1#Bits and qubits: An Introduction|1.3]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Circuit Diagrams for Qubit Gates|2.3.1]], [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]], [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 8 - Noise in Quantum Systems#Modelling Open System Evolution|8.3]], [[Chapter 8 - Noise in Quantum Systems#Fixed-Basis Operations|8.3.2]], [[Chapter 8 - Noise in Quantum Systems#Unitary Freedom|8.4.1]], [[Chapter 8 - Noise in Quantum Systems#Physical Interpretation of the Unitary Freedom|8.4.2]], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']], [[Appendix D - Group Theory#Introduction|'''D.1''']]<br />
::active [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
::passive [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=890Index2010-05-03T00:22:40Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:Dirac notation [[Appendix C - Vectors and Linear Algebra#Introduction|'''C.2.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transformation [[Chapter 1#Bits and qubits: An Introduction|1.3]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Circuit Diagrams for Qubit Gates|2.3.1]], [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]], [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]], <br />
::active<br />
::passive<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=889Index2010-05-03T00:17:00Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:Dirac notation [[Appendix C - Vectors and Linear Algebra#Introduction|'''C.2.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transformation [[Chapter 1#Bits and qubits: An Introduction|1.3]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Circuit Diagrams for Qubit Gates|2.3.1]], [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]], [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
::active<br />
::passive<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=888Index2010-05-03T00:14:30Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:Dirac notation [[Appendix C - Vectors and Linear Algebra#Introduction|'''C.2.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transformation [[Chapter 1#Bits and qubits: An Introduction|1.3]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Circuit Diagrams for Qubit Gates|2.3.1]], [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]], [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Chapter 2 - Qubits and Collections of Qubits#Projection Operators|2.7.2]]<br />
::active<br />
::passive<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=887Index2010-05-02T23:52:03Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:Dirac notation [[Appendix C - Vectors and Linear Algebra#Introduction|'''C.2.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transformation <br />
::active<br />
::passive<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_A_-_Basic_Probability_Concepts&diff=886Appendix A - Basic Probability Concepts2010-05-02T23:45:08Z<p>Rceballos: </p>
<hr />
<div>In this appendix definitions and some example calculations are<br />
presented which will aid in our discussions. This is not meant to be<br />
a comprehensive introduction to the topic. It is primarily meant to<br />
serve as a means for introducing notation and terminology for the<br />
course. (This example is a variation of one given by David Griffiths<br />
in ''Intoduction to Quantum Mechanics'' ([[Bibliography#Griffiths:qmbook|David J. Giffiths’s book]]) <br />
<br />
''Example'': Suppose that in some room, there are four people. Their heights in meters are: <br />
#1 person is 1.5 meters tall<br />
#1 person is 1.6 meters tall<br />
#2 people are 1.8 meters tall<br />
We might write this as (<math>N\,\!</math> will stand for the number of people) <br />
<math>N(1.5) = 1\,\!</math>, <math>N(1.6)=1\,\!</math>, <math>N(1.8)=2\,\!</math> and the total number of people is <br />
<center><math><br />
N = \sum_{j=0}^\infty N(j),<br />
\,\!</math></center><br />
where <math>j\,\!</math> runs over all values and in this case we are rounding to the<br />
nearest tenth of a meter. Here <math>N\,\!</math> =4 of course. <br />
<br />
Now if I draw a name out of a hat which contains each persons name<br />
once, I will get the persons name which is 1.6 meters tall with<br />
probability <math>1/4\,\!</math>. (We assume that each person has a unique name and<br />
that it appears once and only once in the hat.) We write this as<br />
<center><math><br />
P(1.6) = 1/4,<br />
\,\!</math></center><br />
and we would generally write for any value<br />
<center><math><br />
P(j) = \frac{N(j)}{N}. <br />
\,\!</math></center><br />
Now since we are going to get someone's name when we draw, we must<br />
have <br />
<center><math><br />
\sum_j P(j) = 1,<br />
\,\!</math></center><br />
which is easy enough to check. <br />
<br />
There are several aspects of this probability distribution that we might like to know. Here are some which are particularly useful: <!-- \index{median}\index{mean} \index{average}--><br />
#The ''most probable'' values for the height is 1.8 meters.<br />
#The ''median'' is 1.7 meters (two people below, and two above)<br />
#The ''average'' (or ''mean'') is given by<br />
{{Equation|<math>\begin{align}\left\langle height\right\rangle &= \frac{1(1.5)+1(1.6)+2(1.8)}{4} \\ &= \frac{6.8}{4} = 1.7 \end{align}</math>|A.1}} <br />
Note that the mean and the median do not have to be the same. The<br />
median is the middle number in the list, if there is an odd number,<br />
otherwise it is the mean of the two in the middle. These two just<br />
happen to be the same here. <br />
The bracket <math>\left\langle\cdot\right\rangle\,\!</math> is fairly standard notation and it means<br />
generally, that we obtain the ''average value''<!-- \index{average}--> <br />
of a function by calculating <br />
<center><math>\left\langle f(j)\right\rangle = \sum_{j=0}^\infty f(j)P(j).\,\!</math></center><br />
For the average this is just <br />
<center><math><br />
\left\langle j\right\rangle = \sum_{j=0}^\infty jP(j)= \sum_{j=0}^\infty j\frac{N(j)}{N}. \,\!</math></center><br />
<br />
'''Note:''' The ''average value'' is called the ''expectation value'' <!-- \index{expectation value} --> in quantum mechanics. This can be<br />
misleading becase it is ''not'' the most probable and it is not <nowiki>''what to expect.''</nowiki> <br />
<br />
When one would like to discuss a properties of a particular probability distribution, describing it takes some effort. It is not enough to know the average, median, and most probable values. This leaves a lot of details of the probability distribution unknown to us if these are all we are given. What else would one like to know? Without describing it entirely, one may like to know more about the <nowiki>''shape''</nowiki> of the distribution. For example, how spread out is it?<br />
<br />
The most important measure of this is the ''variance'',<!-- \index{variance}--> which is the ''standard deviation'', <!-- \index{standard deviation} --> squared. The variance is defined as (in terms of our variable <math>j\,\!</math>) <br />
{{Equation|<math>\sigma^2 = \langle(\Delta j)^2\rangle, \,\!</math>|A.2}}<br />
where <math>\Delta j = j -\langle j \rangle\,\!</math>. This can also be written as <br />
{{Equation|<math>\sigma^2 = \langle j^2\rangle - \langle j \rangle^2.\,\!</math>|A.3}}</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Appendix_C_-_Vectors_and_Linear_Algebra&diff=885Talk:Appendix C - Vectors and Linear Algebra2010-05-02T23:41:28Z<p>Rceballos: </p>
<hr />
<div><br />
<br />
I think that perhaps one more example of tensor multiplication with 3X3 matrices would be illuminating and helpful. I just wanted to make that suggestion, Russ.</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=884Appendix C - Vectors and Linear Algebra2010-05-02T23:33:07Z<p>Rceballos: /* Example 2 */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by a hat (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors. <!-- \index{basis vectors!real} --> <br />
This is because any vector<br />
(in real three-dimensional space) can be written in terms of these unit/basis vectors. In<br />
some sense they are the basic components of any vector. Other basis<br />
vectors could be use, however. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in higher dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will briefly discuss several of these aspects here,<br />
however, we will first provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, then<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector and in Dirac notation, the symbol <math>\left\langle\cdot \right\vert\!</math>, is called a ''bra''. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and <br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C.16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>. (The <math> \left\langle\phi\mid\psi\right\rangle \,\!</math><br />
is called a ''bracket'' which is the product of the ''bra'' and the ''ket''.) The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, <br />
often when one vector is transformed to another, the transformation can be viewed as a transformation of the components of the vector and is <br />
also represented by a matrix. Thus transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
In the case that we consider a matrix transformation of basis elements, we call it a passive transformation. (The transformation does nothing to the object, but only changes the basis in which the object is described.) An active transformation is one where the object itself is transformed. Often these two transformations, active and passive, are very simply related. However, the distinction can be very important. <br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
<br />
====Example 1====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
====Example 2====<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']] are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
====Example 3====<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right) <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=832Appendix C - Vectors and Linear Algebra2010-04-14T20:20:41Z<p>Rceballos: /* Linear Algebra: Matrices */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by a hat (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors. <!-- \index{basis vectors!real} --> <br />
This is because any vector<br />
(in real three-dimensional space) can be written in terms of these unit/basis vectors. In<br />
some sense they are the basic components of any vector. Other basis<br />
vectors could be use, however. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in higher dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will briefly discuss several of these aspects here,<br />
however, we will first provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, the<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector and in Dirac notation , the symbol <math>\left\langle\cdot \right\vert\!</math>, called a ''bra''. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and<br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C.16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>. (The <math> \left\langle\phi\mid\psi\right\rangle \,\!</math><br />
is called a ''bracket'' which is the product of the ''bra'' and the ''ket''.) The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, most<br />
often when one vector is transformed to another, the transformation is<br />
represented by a matrix. Such transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
<br />
====Example 1====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
====Example 2====<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues \index{eigenvalues} are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
====Example 3====<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right) <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Appendix_C_-_Vectors_and_Linear_Algebra&diff=831Talk:Appendix C - Vectors and Linear Algebra2010-04-14T20:18:00Z<p>Rceballos: </p>
<hr />
<div>[[Chapter 2 - Qubits and Collections of Qubits#eq2.1|2.1]]<br />
<br />
The "bra" notation used to define the Hermitian conjugate of the "ket" vector, was backwards I believe. It was the same as when you defined the "ket" vector earlier. I think I fixed it adequately. Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Index&diff=830Talk:Index2010-04-14T20:15:55Z<p>Rceballos: </p>
<hr />
<div>I'm not finding a term in the text for "partial trace operator," "maximally mixed state, two qubits," "mean," "projection operator," or "scalability." Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=829Index2010-04-14T20:15:00Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']], [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=828Appendix C - Vectors and Linear Algebra2010-04-14T20:13:19Z<p>Rceballos: /* More Dirac Notation */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by a hat (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors. <!-- \index{basis vectors!real} --> <br />
This is because any vector<br />
(in real three-dimensional space) can be written in terms of these unit/basis vectors. In<br />
some sense they are the basic components of any vector. Other basis<br />
vectors could be use, however. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in higher dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will discuss (briefly) several of these here,<br />
but first we will provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, the<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector and in Dirac notation , the symbol <math>\left\langle\cdot \right\vert\!</math>, called a ''bra''. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and<br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C.16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>. (The <math> \left\langle\phi\mid\psi\right\rangle \,\!</math><br />
is called a ''bracket'' which is the product of the ''bra'' and the ''ket''.) The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, most<br />
often when one vector is transformed to another, the transformation is<br />
represented by a matrix. Such transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
<br />
====Example 1====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
====Example 2====<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues \index{eigenvalues} are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
====Example 3====<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right) <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Appendix_C_-_Vectors_and_Linear_Algebra&diff=827Talk:Appendix C - Vectors and Linear Algebra2010-04-14T20:04:36Z<p>Rceballos: </p>
<hr />
<div>[[Chapter 2 - Qubits and Collections of Qubits#eq2.1|2.1]]<br />
<br />
The "bra" notation used to define the Hermitian conjugate of the "ket" vector, is backwards I believe. It is the same as when you defined the "ket" vector earlier. Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=826Appendix C - Vectors and Linear Algebra2010-04-14T19:58:30Z<p>Rceballos: /* Real Vectors */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by a hat (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors. <!-- \index{basis vectors!real} --> <br />
This is because any vector<br />
(in real three-dimensional space) can be written in terms of these unit/basis vectors. In<br />
some sense they are the basic components of any vector. Other basis<br />
vectors could be use, however. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in higher dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will discuss (briefly) several of these here,<br />
but first we will provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, the<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector and in Dirac notation , the symbol <math>\left\vert \cdot \right\rangle\,\!</math>, called a ''bra''. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and<br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C.16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>. (The <math> \left\langle\phi\mid\psi\right\rangle \,\!</math><br />
is called a ''bracket'' which is the product of the ''bra'' and the ''ket''.) The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, most<br />
often when one vector is transformed to another, the transformation is<br />
represented by a matrix. Such transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
<br />
====Example 1====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
====Example 2====<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues \index{eigenvalues} are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
====Example 3====<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right) <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=817Appendix C - Vectors and Linear Algebra2010-04-12T16:50:19Z<p>Rceballos: /* Real Vectors */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by the hats (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors. <!-- \index{basis vectors!real} --> <br />
This is because any vector<br />
(in real three-dimensional space) can be written in terms of these unit/basis vectors. In<br />
some sense they are the basic components of any vector. Other basis<br />
vectors could be use, however. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in higher dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will discuss (briefly) several of these here,<br />
but first we will provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, the<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and<br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>.<br />
The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, most<br />
often when one vector is transformed to another, the transformation is<br />
represented by a matrix. Such transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
====Examples====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues \index{eigenvalues} are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=816Appendix C - Vectors and Linear Algebra2010-04-12T16:48:41Z<p>Rceballos: /* Real Vectors */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by the hats (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors. <!-- \index{basis vectors!real} --> <br />
This is because any vector<br />
(in real three-dimensional space) can be written in terms of these unit/basis vectors. In<br />
some sense they are basic components of any vector. Other basis<br />
vectors could be used though. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in more dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will discuss (briefly) several of these here,<br />
but first we will provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, the<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and<br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>.<br />
The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, most<br />
often when one vector is transformed to another, the transformation is<br />
represented by a matrix. Such transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
====Examples====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues \index{eigenvalues} are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=815Appendix C - Vectors and Linear Algebra2010-04-12T16:46:57Z<p>Rceballos: /* Real Vectors */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector, and the components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. The unit vectors, denoted by the hats (<math>\hat{\cdot}\,\!</math>) are also<br />
known as ''basis'' vectors, <!-- \index{basis vectors!real} --> <br />
because any vector<br />
(in real three-dimensional space) can be written in terms of them. In<br />
some sense they are basic components of any vector. Other basis<br />
vectors could be used though. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in more dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will discuss (briefly) several of these here,<br />
but first we will provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, the<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and<br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>.<br />
The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, most<br />
often when one vector is transformed to another, the transformation is<br />
represented by a matrix. Such transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
====Examples====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues \index{eigenvalues} are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&diff=814Appendix C - Vectors and Linear Algebra2010-04-12T16:41:14Z<p>Rceballos: /* Real Vectors */</p>
<hr />
<div>===Introduction===<br />
<br />
This appendix introduces some aspects of linear algebra and complex<br />
algebra which will be helpful for the course. In addition, Dirac<br />
notation is introduced and explained.<br />
<br />
===Vectors===<br />
<br />
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics. <br />
<br />
====Real Vectors====<br />
<br />
<br />
The simple definition of a vector - an object which has magnitude and<br />
direction - is helpful to keep in mind even when dealing with complex<br />
and/or abstract vectors as we will here. In three dimensional space,<br />
a vector is often written as<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},<br />
</math></center><br />
where the hat (<math>\hat{\cdot}\,\!</math>) denotes a unit vector. The components<br />
<math>v_i\,\!</math>, <math>i = x,y,z\,\!</math> are just numbers. These unit vectors are also<br />
called ''basis'' vectors <!-- \index{basis vectors!real} --> <br />
because any vector<br />
(in real three-dimensional space) can be written in terms of them. In<br />
some sense they are basic components of any vector. Other basis<br />
vectors could be used though. Some other common choices are those that<br />
are used for spherical and cylindrical<br />
coordinates. When dealing with more abstract and/or complex vectors,<br />
it is often helpful to ask what one would do for an ordinary<br />
three-dimensional vector. For example, properties of unit vectors,<br />
dot products, etc. in three-dimensions are similar to the analogous<br />
constructions in more dimensions. <br />
<br />
The ''inner product'',<!-- \index{inner product}--> or ''dot product''<!--\index{dot product}--> for two real three-dimensional vectors<br />
<center><math><br />
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; <br />
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},<br />
</math></center><br />
can be computed as follows<br />
<center><math><br />
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.<br />
</math></center><br />
For the inner product of <math>\vec{v}\,\!</math> with itself, we get the square of<br />
the magnitude of <math>\vec{v}\,\!</math> denoted <math>|\vec{v}|^2\,\!</math>,<br />
<center><math><br />
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +<br />
v_zv_z=v_x^2+v_y^2+v_z^2. <br />
</math></center><br />
If we want a unit vector in the direction of <math>\vec{v}\,\!</math>, then we divide<br />
by its magnitude to get a unit vector,<br />
<center><math><br />
\hat{v} = \frac{\vec{v}}{|\vec{v}|}. <br />
</math></center><br />
Now, of course, <math>\hat{v}\cdot\hat{v}= 1\,\!</math> as can easily be checked. <br />
<br />
There are several ways to represent a vector. The ones we will use<br />
most often are column and row vector notations. So, for example, we<br />
could write the vector above as<br />
<center><math><br />
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
In this case, our unit vectors are represented by the following <br />
<center><math><br />
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0<br />
\end{array}\right), \;\; <br />
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0<br />
\end{array}\right), \;\;<br />
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1<br />
\end{array}\right).<br />
\,\!</math></center><br />
<br />
We next turn to the subject of complex vectors along with the relevant<br />
notation. <br />
We will see how to compute the inner product later, but some other<br />
definitions will be required.<br />
<br />
====Complex Vectors====<br />
<br />
For complex vectors in quantum mechanics, Dirac notation is most often<br />
used. This notation uses a <math>\left\vert \cdot \right\rangle\,\!</math>, <br />
called a ''ket'', for a vector, so our vector <math>\vec{v}\,\!</math> would be<br />
<center><math><br />
\left\vert v \right\rangle = \left(\begin{array}{c} v_x \\ v_y \\ v_z<br />
\end{array}\right). <br />
</math></center><br />
<br />
For qubits, i.e. two-state quantum systems, will often be written as<br />
complex vectors <br />
{{Equation|<math>\begin{align} \left\vert \psi \right\rangle &= \left(\begin{array}{c} \alpha \\ \beta <br />
\end{array}\right) \\<br />
&=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}</math>|C.1}} <br />
where<br />
<center><math><br />
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 <br />
\end{array}\right), \;\;\mbox{and} \;\;<br />
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 <br />
\end{array}\right)<br />
</math></center><br />
are the basis vectors. The two numbers <math>\alpha\,\!</math> and<br />
<math>\beta\,\!</math> are complex numbers, so the vector is said to<br />
be a complex vector.<br />
<br />
<br />
<br />
===Linear Algebra: Matrices===<br />
<br />
<br />
There are many aspects of linear algebra that are quite useful in<br />
quantum mechanics. We will discuss (briefly) several of these here,<br />
but first we will provide some definitions and properties which will<br />
be useful as well as fixing notation. Some familiarity with matrices<br />
will be assumed, but many basic defintions are also included. <br />
<br />
<br />
Let us denote some <math>m\times n\,\!</math> matrix by <math>A\,\!</math>. The set of all <math>m\times<br />
n\,\!</math> matrices with real entries is <math>M(n\times m,\mathbb{R})\,\!</math>. Such matrices<br />
are said to be real since they have all real entries. Similarly, the<br />
set of <math>m\times n\,\!</math> complex matrices is <math>M(m\times n,\mathbb{C})\,\!</math>. For the<br />
set of set of square <math>n\times n\,\!</math> complex matrices, we simply write<br />
<math>M(n,\mathbb{C})\,\!</math>. <br />
<br />
<br />
We will also refer to the set of matrix elements, <math>a_{ij}\,\!</math> where the<br />
first index (<math>i\,\!</math> in this case) labels the row and the second <math>(j)\,\!</math><br />
labels the column. Thus the element <math>a_{23}\,\!</math> is the element in the<br />
second row and third column. A comma is inserted if there is some<br />
ambiguity. For example, in a large matrix the element in the<br />
2nd row and 12th<br />
column is written as <math>a_{2,12}\,\!</math> to distinguish between the<br />
21st row and 2nd column. <br />
<br />
<br />
<br />
====Complex Conjugate====<br />
<br />
The ''complex conjugate of a matrix'' <!-- \index{complex conjugate!of a matrix}--><br />
is the matrix with each element replaced by its complex conjugate. In<br />
other words, to take the complex conjugate of a matrix, one takes the<br />
complex conjugate of each entry in the matrix. We denote the complex<br />
conjugate with a ``star,'' e.g. <math>A^*\,\!</math>. For example,<br />
{{Equation|<math>\begin{align}<br />
A^* &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^* \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11}^* & a_{12}^* & a_{13}^* \\<br />
a_{21}^* & a_{22}^* & a_{23}^* \\<br />
a_{31}^* & a_{32}^* & a_{33}^* \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.2}}<br />
(Notice that the notation for a matrix is a capital letter, whereas<br />
the entries are numbers, so they are represented by lower case<br />
letters.)<br />
<br />
====Transpose====<br />
<br />
<br />
The ''transpose'' <!-- \index{transpose} --> of a matrix is the same set of<br />
elements but the first row becomes the first column, the second row<br />
becomes the second column, etc. Thus the rows and columns are<br />
interchanged. For example, for a square <math>3\times 3\,\!</math> matrix, the<br />
transpose is given by<br />
{{Equation|<math>\begin{align}<br />
A^T &=& \left(\begin{array}{ccc}<br />
a_{11} & a_{12} & a_{13} \\<br />
a_{21} & a_{22} & a_{23} \\<br />
a_{31} & a_{32} & a_{33} \end{array}\right)^T \\<br />
&=& \left(\begin{array}{ccc}<br />
a_{11} & a_{21} & a_{31} \\<br />
a_{12} & a_{22} & a_{32} \\<br />
a_{13} & a_{23} & a_{33} \end{array}\right). <br />
\end{align}<br />
\,\!</math>|C.3}}<br />
<br />
<br />
<br />
====Hermitian Conjugate====<br />
<br />
<br />
The complex conjugate and transpose is called the ''Hermitian conjugate'', or simply the ''dagger'' of a matrix. It is called the dagger, because the symbol used to denote it is a dagger<br />
(<math>\dagger\,\!</math>), viz. <br />
{{Equation|<math><br />
(A^T)^* = (A^*)^T \equiv A^\dagger.<br />
\,\!</math>|C.4}}<br />
For our <math>3\times 3\,\!</math> example, <br />
<center><math><br />
A^\dagger = \left(\begin{array}{ccc}<br />
a_{11}^* & a_{21}^* & a_{31}^* \\<br />
a_{12}^* & a_{22}^* & a_{32}^* \\<br />
a_{13}^* & a_{23}^* & a_{33}^* \end{array}\right). <br />
\,\!</math></center><br />
If a matrix is its own Hermitian conjugate, i.e. <math>A^\dagger = A\,\!</math>, the<br />
we call it a ''Hermitian matrix''. <!-- \index{Hermitian matrix}--><br />
(Clearly this is only possible for square matrices.) Hermitian<br />
matrices are very important in quantum mechanics since their<br />
eigenvalues are real. (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)<br />
<br />
====Index Notation====<br />
<br />
Very often we write the product of two matrices <math>A\,\!</math> and <math>B\,\!</math> simply as<br />
<math>AB\,\!</math> and let <math>C=AB\,\!</math>. However, it is also quite useful to write this<br />
in component form. In this case, if these are <math>n\times n\,\!</math> matrices <br />
<center><math><br />
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.<br />
\,\!</math></center><br />
This says that the element in the <math>i^{\mbox{th}}\,\!</math> row and<br />
<math>j^{\mbox{th}}\,\!</math> column of the matrix <math>C\,\!</math> is the sum <math>\sum_1^n<br />
a_{ij}b_{jk}\,\!</math>. The transpose of <math>C\,\!</math> has elements<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.<br />
\,\!</math></center><br />
Now if we were to transpose <math>A\,\!</math> and <math>B\,\!</math> as well, this would read<br />
<center><math><br />
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_1^n b^T_{ij} a^T_{jk}.<br />
\,\!</math></center><br />
This gives us a way of seeing the general rule that <br />
<center><math><br />
C^T = B^TA^T. <br />
\,\!</math></center><br />
It follows that <br />
<center><math><br />
C^\dagger = B^\dagger A^\dagger.<br />
\,\!</math></center><br />
<br />
<br />
<br />
<br />
====The Trace====<br />
<br />
The ''trace'' <!-- \index{trace}--> of a matrix is the sum of the diagonal<br />
elements and is denoted <math>\mbox{Tr}\,\!</math>. So for example, the trace of an<br />
<math>n\times n\,\!</math> matrix <math>A\,\!</math> is <br />
<center><math><br />
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}<br />
\,\!</math></center><br />
<br />
Some useful properties of the trace are the following:<br />
#<math>\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!</math><br />
#<math>\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!</math><br />
Using the first of these results,<br />
<center><math><br />
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).<br />
\,\!</math></center><br />
This relation is used so often that we state it here explicitly. <br />
<br />
<br />
<br />
====The Determinant====<br />
<br />
For a square matrix, the determinant is quite a useful thing. For<br />
example, an <math>n\times n\,\!</math> matrix is invertible if and only if its<br />
determinant is not zero. So let us define the determinant and give<br />
some properties and examples. <br />
<br />
<br />
The ''determinant''<!--\index{determinant}--> of a <math>2\times 2\,\!</math> matrix <br />
{{Equation|<math><br />
N = \left(\begin{array}{cc}<br />
a & b \\<br />
c & d \end{array}\right),<br />
\,\!</math>|C.5}}<br />
is given by<br />
{{Equation|<math><br />
\det(N) = ad-bc.<br />
\,\!</math>|C.6}}<br />
Higher-order determinants can be written in terms of smaller ones in<br />
the standard way. <br />
<br />
<br />
The ''determinant''<!-- \index{determinant}--> of a matrix <math>A\,\!</math> can be<br />
also be written in terms of its components as <br />
{{Equation|<math><br />
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}<br />
a_{1i}a_{2j}a_{3k}a_{4l} ...,<br />
\,\!</math>|C.7}}<br />
where the symbol <br />
{{Equation|<math><br />
\epsilon_{ijkl...} = \begin{cases}<br />
+1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\<br />
-1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.8}}<br />
<br />
Let us consider the example of the <math>3\times 3\,\!</math> matrix <math>A\,\!</math> given<br />
above. The determinant can be calculated by<br />
<center><math><br />
\det(A) = \sum_{i,j,k} \epsilon_{ijk}<br />
a_{1i}a_{2j}a_{3k},<br />
\,\!</math></center><br />
where, explicitly, <br />
{{Equation|<math><br />
\epsilon_{ijk} = \begin{cases}<br />
+1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\<br />
-1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\<br />
\;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).<br />
\end{cases}<br />
\,\!</math>|C.9}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\det(A) &=& \epsilon_{123}a_{11}a_{22}a_{33} <br />
+\epsilon_{132}a_{11}a_{23}a_{32}<br />
+\epsilon_{231}a_{12}a_{23}a_{31} \\<br />
&&+\epsilon_{213}a_{12}a_{21}a_{33}<br />
+\epsilon_{312}a_{13}a_{21}a_{32}<br />
+\epsilon_{213}a_{13}a_{21}a_{32}.<br />
\end{align}<br />
\,\!</math>|C.10}}<br />
Now given the values of <math>\epsilon_{ijk}\,\!</math> in Eq.~(\ref{eq:3depsilon}),<br />
this is<br />
<center><math><br />
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} <br />
- a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{21}a_{32}.<br />
\,\!</math></center><br />
<br />
The determinant has several properties which are useful to know. A<br />
few are listed here. <br />
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: <center><math> \det(A) = \det(A^T).\,\!</math></center><br />
#The determinant of a product is the product of determinants: <center><math> \det(AB) = \det(A)\det(B).\,\!</math></center><br />
From this last property, another specific property can be derived.<br />
Suppose we take the determinant of the product of a matrix and its<br />
inverse we find<br />
<center><math><br />
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,<br />
\,\!</math></center><br />
since the determinant of the identity is one. This implies that<br />
<center><math><br />
\det(U^{-1}) = \frac{1}{\det(U)}. <br />
\,\!</math></center><br />
<br />
====The Inverse of a Matrix====<br />
<br />
<br />
The inverse <!-- \index{inverse}--> of a square matrix <math>A\,\!</math> is another matrix,<br />
denoted <math>A^{-1}\,\!</math> such that <br />
<center><math><br />
AA^{-1} = A^{-1}A = \mathbb{I},<br />
\,\!</math></center><br />
where <math>\mathbb{I}\,\!</math> is the identity matrix consisting of zeroes everywhere<br />
except the diagonal which has ones. For example the <math>3\times 3\,\!</math><br />
identity matrix is <br />
<center><math><br />
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br />
\end{array}\right). <br />
\,\!</math></center><br />
<br />
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.'' Thus one only needsd to calculate the<br />
determinant to see if a matrix has an inverse or not.<br />
<br />
====Unitary Matrices====<br />
<br />
A ''unitary matrix'' <!-- \index{unitary matrix} --> <math>U\,\!</math> is one whose<br />
inverse is also its Hermitian conjugate, <math>U^\dagger = U^{-1}\,\!</math> so that <br />
<center><math><br />
U^\dagger U = UU^\dagger = \mathbb{I}.<br />
\,\!</math></center><br />
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.<!-- \index{special unitary matrix}--> The set of<br />
<math>n\times n\,\!</math> unitary matrices is denoted<br />
U<math>(n)\,\!</math> and the set of special unitary matrices is denoted SU<math>(n)\,\!</math>. <br />
<br />
Unitary matrices are particularly important in quantum mechanics<br />
because they describe the evolution, or change, of quantum states.<br />
They are able to do this because unitary matrices have the property that rows and<br />
columns, viewed as vectors, are orthonormal. (To see this, an example<br />
is provided below.) This means that when<br />
they act on a basis vector of the form (one 1, in say the <math>j</math>th spot, and zeroes everywhere else)<br />
{{Equation|<math> \left\vert j\right\rangle = <br />
\left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 <br />
\end{array}\right), <br />
\,\!</math>|C.11}}<br />
the result is a normalized complex vector. Acting on a set of<br />
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]],<br />
will produce another orthonormal set. <br />
<br />
Let us consider the example of a <math>2\times 2\,\!</math> unitary matrix, <br />
{{Equation|<math><br />
U = \left(\begin{array}{cc} <br />
a & b \\ <br />
c & d <br />
\end{array}\right).<br />
\,\!</math>|C.12}}<br />
The inverse of this matrix is the Hermitian conjugate, so the inverse<br />
is given by<br />
{{Equation|<math><br />
U ^{-1} = U^\dagger = \left(\begin{array}{cc} <br />
a^* & c^* \\ <br />
b^* & d^* <br />
\end{array}\right),<br />
\,\!</math>|C.13}}<br />
provided that the matrix <math>U\,\!</math> satisfies the constraints<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |b|^2 = 1, \; & \; ac^*+bd^* =0 \\<br />
ca^*+db^*=0, \; & \; |c|^2 + |d|^2 =1,<br />
\end{align}\,\!</math>|C.14}}<br />
and<br />
{{Equation|<math>\begin{align}<br />
|a|^2 + |c|^2 = 1, \; & \; ba^*+dc^* =0 \\<br />
b^*a+d^*c=0, \; & \; |b|^2 + |d|^2 =1.<br />
\end{align}\,\!</math>|C.15}}<br />
Looking at each row as a vector, the constraints in<br />
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the<br />
vectors forming the rows. Similarly, the constraints in<br />
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the<br />
vectors forming the columns when viewed as vectors.<br />
<br />
===More Dirac Notation===<br />
<br />
<br />
Now that we have a definition of Hermitian conjugate, we consider the<br />
case for a <math>1\times n\,\!</math> matrix, i.e. a vector. In Dirac notation, we<br />
had <br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right),<br />
\,\!</math></center><br />
So the Hermitian conjugate comes up so often that we use the following<br />
notation for vectors,<br />
<center><math><br />
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\<br />
\beta \end{array}\right)^\dagger <br />
= \left( \alpha^*, \; \beta^* \right). <br />
\,\!</math></center><br />
This is a row vector. Let us consider a second complex vector <br />
<center><math><br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
The ''inner product'' <!-- \index{inner product}--> between <math>\left\vert\psi\right\rangle\,\!</math> and<br />
<math>\left\vert\phi\right\rangle\,\!</math> is computed as follows:<br />
{{Equation|<math>\begin{align}<br />
\left\langle\phi\mid\psi\right\rangle & \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle \\<br />
&= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) \\<br />
&= \gamma^*\alpha + \delta^*\beta.<br />
\end{align}<br />
</math>|C16}}<br />
If these two vectors are ''orthogonal'', <!-- \index{orthogonal!vectors} --><br />
then their inner product is zero, <math>\left\langle\phi\mid\psi\right\rangle =0\,\!</math>.<br />
The inner product of <math>\left\vert\psi\right\rangle\,\!</math> with itself is <br />
<center><math><br />
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.<br />
\,\!</math></center><br />
If this vector is normalized then <math>\left\langle\psi\mid\psi\right\rangle = 1\,\!</math>. <br />
<br />
More generally, we will consider vectors in <math>N\,\!</math> dimensions. In this<br />
case we write the vector in terms of a set of basis vectors<br />
<math>\{\left\vert i\right\rangle\}\,\!</math>, where <math>i = 0,1,2,...N-1\,\!</math>. This is an ordered set of<br />
vectors which are just labeled by integers. If the set is orthogonal,<br />
then <br />
<center><math><br />
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j,<br />
\,\!</math></center><br />
and if they are normalized, then <br />
<center><math><br />
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.<br />
\,\!</math></center><br />
If both of these are true, i.e., the entire set is orthonormal, we can<br />
write,<br />
<center><math><br />
\left\langle i\mid j\right\rangle = \delta_{ij},<br />
\,\!</math></center><br />
where the symbol <math>\delta_{ij}\,\!</math> is called the Kronecker delta <!-- \index{Kronecker delta} --> and is defined by <br />
{{Equation|<math><br />
\delta_{ij} = \begin{cases}<br />
1, & \mbox{if } i=j, \\<br />
0, & \mbox{if } i\neq j.<br />
\end{cases}<br />
</math>|C.17}}<br />
Now consider <math>(N+1)\,\!</math>-dimensional vectors by letting two such vectors<br />
be expressed in the same basis as<br />
<center><math><br />
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle,<br />
\,\!</math></center><br />
and<br />
<center><math><br />
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle. <br />
\,\!</math></center><br />
Then the inner product <!--\index{inner product}--> is<br />
{{Equation|<math>\begin{align}<br />
\left\langle\Psi\mid\Phi\right\rangle &= \left(\sum_{i=0}^{N}<br />
\alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right) \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle \\<br />
&= \sum_{ij} \alpha_i^*\beta_j\delta_{ij} \\<br />
&= \sum_i\alpha^*_i\beta_i,<br />
\end{align}<br />
</math>|C.18}}<br />
where we have used the fact that the delta function is zero unless<br />
<math>i=j\,\!</math> to get the last equality. For the inner product of a vector<br />
with itself, we get<br />
<center><math><br />
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.<br />
\,\!</math></center><br />
This immediately gives us a very important property of the inner<br />
product. It tells us that in general,<br />
<center><math><br />
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0<br />
\Leftrightarrow \left\vert\Phi\right\rangle = 0. <br />
\,\!</math></center><br />
(Just in case you don't know, the symbol <math>\Leftrightarrow\,\!</math> means <nowiki>"if and only if"</nowiki> sometimes written as <nowiki>"iff."</nowiki>) <br />
<br />
We could also expand a vector in a different basis. Let us suppose<br />
that the set <math>\{\left\vert e_k \right\rangle\}\,\!</math> is an orthonormal basis <math>(\left\langle e_k \mid e_l\right\rangle =<br />
\delta_{kl})\,\!</math> which is different from the one considered earlier. We<br />
could expand our vector <math>\left\vert\Psi\right\rangle\,\!</math> in terms of our new basis by<br />
expanding our new basis in terms of our old basis. Let us first<br />
expand the <math>\left\vert e_k\right\rangle\,\!</math> in terms of the <math>\left\vert j\right\rangle\,\!</math>:<br />
{{Equation|<math><br />
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,<br />
</math>|C.19}}<br />
so that <br />
{{Equation|<math>\begin{align}<br />
\left\vert \Psi\right\rangle &= \sum_j \alpha_j\left\vert j\right\rangle \\<br />
&= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle \\ <br />
&= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, <br />
\end{align}<br />
</math>|C.20}}<br />
where <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. <br />
</math>|C.21}}<br />
Notice that the insertion of <math>\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!</math> didn't do anything to our original vector. It is the same vector, just in a<br />
different basis. Therefore, this is effectively the identity operator<br />
<center><math><br />
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert. <br />
\,\!</math></center><br />
This is an important and quite useful relation. <br />
Now, to interpret Eq.[[#eqC.19|(C.19)]], we can draw a close<br />
analogy with three-dimensional real vectors. The inner product<br />
<math>\left\vert e_k \right\rangle \left\langle j \right\vert\,\!</math> can be interpreted as the projection of one vector onto<br />
another. This provides the part of <math>\left\vert j \right\rangle\,\!</math> along <math>\left\vert e_k \right\rangle\,\!</math>.<br />
<br />
===Transformations===<br />
<br />
Suppose we have two different orthogonal bases, <math>\{e_k\}\,\!</math>, <math>\{j\}\,\!</math>.<br />
The numbers <math>\left\langle e_k\mid j\right\rangle\,\!</math> for all the different <math>k\,\!</math> and <math>j\,\!</math> are<br />
often referred to as matrix elements since the set forms a matrix with<br />
<math>k\,\!</math> labelling the rows, and <math>j\,\!</math> labelling the columns. Therefore, we<br />
can write the transformation from one basis to another with a matrix<br />
transformation. Let <math>M\,\!</math> be the matrix with elements <math>m_{kj} =<br />
\left\langle e_k\mid j\right\rangle\,\!</math>. Then the transformation from one basis to another,<br />
written in terms of the coefficients of <math>\left\vert\Psi\right\rangle\,\!</math>, is <br />
{{Equation|<math> A^\prime = MA, </math>|C.22}}<br />
where <br />
<center><math><br />
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\<br />
\alpha_n^\prime \end{array}\right), \;\; <br />
\mbox{ and } \;\;<br />
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\<br />
\alpha_n\end{array}\right).<br />
\,\!</math></center><br />
This sort of transformation is a change of basis. However, most<br />
often when one vector is transformed to another, the transformation is<br />
represented by a matrix. Such transformations can either be<br />
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components <br />
{{Equation|<math><br />
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. <br />
</math>|C.23}}<br />
<br />
For a general transformation matrix <math>T\,\!</math>, acting on a vector,<br />
the matrix elements in a particular basis <math>\left\vert i\right\rangle\,\!</math> are <br />
<center><math><br />
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, <br />
\,\!</math></center><br />
just as elements of a vector can be found using<br />
<center><math><br />
\left\langle i\mid \Psi \right\rangle = \alpha_i.<br />
\,\!</math></center><br />
<br />
A ''similarity transformation'' <!--\index{similarity transformation}--> <br />
of an <math>n\times n\,\!</math> matrix <math>A\,\!</math> by an invertible matrix <math>S\,\!</math> is <math>S A S^{-1}\,\!</math>.<br />
There are (at least) two important things to note about similarity<br />
transformations, <br />
#Similarity transformations leave determinants unchanged. (We say the determinant is invariant under similarity transformations.) This is because <center><math> \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!</math></center><br />
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged. Let <math>A^\prime = SAS^{-1}\,\!</math>, <math>B^\prime = SBS^{-1}\,\!</math>, and <math>C^\prime = SCS^{-1}\,\!</math>. If <math>AB=C\,\!</math>, then <math>A^\prime B^\prime = C^\prime\,\!</math> since <math>A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} = SCS^{-1}=C^\prime\,\!</math>. The two matrices <math>A^\prime\,\!</math> and <math>A\,\!</math> are said to be ''similar''.<br />
<!-- \index{similar matrices} --><br />
<br />
===Eigenvalues and Eigenvectors===<br />
<br />
<!-- \index{eigenvalues}\index{eigenvectors} --><br />
<br />
<br />
<br />
A matrix can always be diagonalized. By this, it is meant that for<br />
every complex matrix <math>M\,\!</math> there is a diagonal matrix <math>D\,\!</math> such that <br />
{{Equation|<math><br />
M = UDV, <br />
\,\!</math>|C.24}}<br />
where <math>U\,\!</math> and <math>V\,\!</math> are unitary matrices. This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix <math>D\,\!</math> are called the ''singular values'' <!--\index{singular values}--> <br />
of the matrix <math>M\,\!</math>. However, the singular values are not always easy to find. <br />
<br />
For the special case that the matrix <math>M\,\!</math> is Hermitian <math>(M^\dagger = M)\,\!</math>, <br />
the matrix <math>M\,\!</math> can be written as<br />
{{Equation|<math><br />
M = U D U^\dagger,<br />
</math>|C.25}}<br />
where <math>U\,\!</math> is unitary <math>(U^{-1}=U^\dagger)\,\!</math>. In this case the elements<br />
of the matrix <math>D\,\!</math> are called ''eigenvalues''. <!--\index{eigenvalues}--><br />
Very often eigenvalues are introduced as solutions to the equation<br />
<center><math><br />
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle<br />
\,\!</math></center><br />
where <math>\left\vert v\right\rangle\,\!</math> a vector called an ''eigenvector''. <!--\index{eigenvector} --><br />
<br />
To find the eigenvalues and eigenvectors of a matrix <math>M\,\!</math>, we follow a<br />
standard procedure which is to calculate the following<br />
{{Equation|<math><br />
\det(\lambda\mathbb{I} - M) = 0,<br />
</math>|C.26}}<br />
and solve for <math>\lambda\,\!</math>. The different solutions for <math>\lambda\,\!</math> is the<br />
set of eigenvalues and this set is called the ''spectrum''. <!-- \index{spectrum}--> Let the different eigenvalues be denoted by <math>\lambda_i\,\!</math>, <math>i=1,2,...,n\,\!</math> fo an <math>n\times n\,\!</math> vector. If two<br />
eigenvalues are equal, we say the spectrum is <br />
''degenerate''. <!--\index{degenerate}--> To find the<br />
eigenvectors, which correspond to different eigenvalues, the equation <br />
<center><math><br />
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle<br />
\,\!</math></center><br />
must be solved for each value of <math>i\,\!</math>. Notice that this equations<br />
holds even if we multiply both sides by some complex number. This<br />
implies that an eigenvector can always be scaled. Usually they are<br />
normalized to obtain an orthonormal set. As we will see by example,<br />
degenerate eigenvalues require some care. <br />
<br />
====Examples====<br />
<br />
Consider a <math>2\times 2\,\!</math> Hermitian matrix<br />
{{Equation|<math><br />
\sigma = \left(\begin{array}{cc} <br />
1+a & b-ic \\<br />
b+ic & 1-a \end{array}\right). <br />
</math>|C.27}}<br />
To find the eigenvalues <!--\index{eigenvalues}--> <br />
of this, we follow a standard procedure which<br />
is to calculate the following<br />
{{Equation|<math><br />
\det(\sigma-\lambda\mathbb{I}) = 0,<br />
</math>|C.28}}<br />
and solve for <math>\lambda\,\!</math>. The eigenvalues of this matrix are given by<br />
<center><math><br />
\det\left(\begin{array}{cc} <br />
1+a-\lambda & b-ic \\<br />
b+ic & 1-a-\lambda \end{array}\right) =0, <br />
\,\!</math></center><br />
which implies the eigenvalues are<br />
<center><math><br />
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}.<br />
\,\!</math></center><br />
and the eigenvectors <!--\index{eigenvectors}--> are<br />
<center><math><br />
v_1=\left(\begin{array}{c}<br />
i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ <br />
2b <br />
\end{array}\right), <br />
v_2= \left(\begin{array}{c}<br />
i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ <br />
2b \end{array}\right).<br />
\,\!</math></center><br />
These expressions are useful for calculating properties of qubit<br />
states as will be seen in the text.<br />
<br />
Now consider a <math>3\times 3\,\!</math> matrix<br />
<center><math><br />
N= \left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right).<br />
\,\!</math></center><br />
First we calculate<br />
<center><math><br />
\det\left(\begin{array}{ccc}<br />
1-\lambda & -i & 0 \\<br />
i & 1-\lambda & 0 \\<br />
0 & 0 & 1-\lambda <br />
\end{array}\right) <br />
= (1-\lambda)[(1-\lambda)^2-1].<br />
\,\!</math></center><br />
This implies that the eigenvalues \index{eigenvalues} are <br />
<center><math><br />
\lambda = 1,0, \mbox{ or } 2.<br />
\,\!</math></center><br />
Let <math>\lambda_1=1\,\!</math>, <math>\lambda_0 = 0\,\!</math>, and <math>\lambda_2 = 2\,\!</math>. <br />
To find eigenvectors, <!--\index{eigenvalues}--> we calculate<br />
{{Equation|<math>\begin{align}<br />
Nv &= \lambda v, \\<br />
\left(\begin{array}{ccc}<br />
1 & -i & 0 \\<br />
i & 1 & 0 \\<br />
0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right) &= \lambda\left(\begin{array}{c} v_1<br />
\\ v_2 \\ v_3 \end{array}\right).<br />
\end{align}<br />
</math>|C.29}}<br />
for each <math>\lambda\,\!</math>. <br />
For <math>\lambda = 1\,\!</math> we get the following equations:<br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= v_1, \\<br />
iv_1+v_2 &= v_2, \\<br />
v_3 &= v_3, <br />
\end{align}<br />
</math>|C.30}}<br />
so <math>v_2 =0\,\!</math>, <math>v_1 =0\,\!</math>, and <math>v_3\,\!</math> is any non-zero number, but we choose<br />
it to normalize the vector. For <math>\lambda<br />
=0\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 &= iv_2, \\<br />
v_3 &= 0, <br />
\end{align}<br />
</math>|C.31}}<br />
and for <math>\lambda = 2\,\!</math>, <br />
{{Equation|<math>\begin{align}<br />
v_1 -iv_2 &= 2v_1, \\<br />
iv_1+v_2 &= 2v_2, \\<br />
v_3 &= 2v_3, <br />
\end{align}<br />
</math>|C.32}}<br />
so that <math>v_1 = -iv_2\,\!</math>. <br />
Therefore, our three eigenvectors <!--\index{eigenvalues}--> are <br />
<center><math><br />
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; <br />
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; <br />
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).<br />
\,\!</math></center><br />
The matrix <br />
<center><math><br />
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}<br />
i/\sqrt{2} & 0 & -i/\sqrt{2} \\<br />
1/\sqrt{2} & 0 & 1/\sqrt{2} \\<br />
0 & 1 & 0 \end{array}\right)<br />
\,\!</math></center><br />
is the matrix that diagonalizes <math>N\,\!</math> in the following way,<br />
<center><math><br />
N = VDV^\dagger,<br />
\,\!</math></center><br />
where<br />
<center><math><br />
D = \left(\begin{array}{ccc}<br />
0 & 0 & 0 \\<br />
0 & 1 & 0 \\<br />
0 & 0 & 2 \end{array}\right)<br />
\,\!</math></center><br />
or, we may write this as<br />
<center><math><br />
V^\dagger N V = D. <br />
\,\!</math></center><br />
This is sometimes called the ''eigenvalue decompostion''<!--\index{eigenvalue decomposition}--> of the matrix and is also written as, <br />
{{Equation|<math><br />
N = \sum_i \lambda_i v_iv^\dagger_i. <br />
\,\!</math>|C.33}}<br />
<br />
(ONE MORE EXAMPLE WITH DEGENERATE E-VALUES)<br />
<br />
===Tensor Products===<br />
<br />
<br />
The tensor product <!--\index{tensor product} --><br />
(also called the Kronecker product) <!--\index{Kronecker product}--><br />
is used extensively in quantum mechanics and<br />
throughout the course. It is commonly denoted with a <math>\otimes\,\!</math><br />
symbol, but this symbol is also often left out. In fact the following<br />
are commonly found in the literature as notation for the tensor<br />
product of two vectors <math>\left\vert\Psi\right\rangle\,\!</math> and <math>\left\vert\Phi\right\rangle\,\!</math><br />
{{Equation|<math>\begin{align}<br />
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle \\<br />
&= \left\vert\Psi\Phi\right\rangle.<br />
\end{align}<br />
</math>|C.34}}<br />
Each of these has its advantages and we will use all of them in<br />
different circumstances. <br />
<br />
The tensor product is also often used for operators. So several<br />
examples <br />
will be given, one which explicitly calculates the tensor product for<br />
two vectors and one which calculates it for two matrices which could<br />
represent operators. However, these are not different in the sense<br />
that a vector is a <math>1\times n\,\!</math> or an <math>n\times 1\,\!</math> matrix. It is also<br />
noteworthy that the two objects in the tensor product need not be of<br />
the same type. In general a tensor product of an <math>n\times m\,\!</math> object<br />
(array) with a <math>p\times q\,\!</math> object will produce an <math>np\times mq\,\!</math><br />
object. <br />
<br />
<br />
In general, the tensor product of two objects is computed as follows.<br />
Let <math>A\,\!</math> be an <math>n\times m\,\!</math> and <math>B\,\!</math> be a <math>p\times q\,\!</math> array <br />
{{Equation|<math><br />
A = \left(\begin{array}{cccc} <br />
a_{11} & a_{12} & \cdots & a_{1m} \\<br />
a_{21} & a_{22} & \cdots & a_{2m} \\<br />
\vdots & & \ddots & \\<br />
a_{n1} & a_{n2} & \cdots & a_{nm} \end{array}\right),<br />
</math>|C.35}}<br />
and similarly for <math>B\,\!</math>. Then <br />
{{Equation|<math><br />
A\otimes B = \left(\begin{array}{cccc} <br />
a_{11}B & a_{12}B & \cdots & a_{1m}B \\<br />
a_{21}B & a_{22}B & \cdots & a_{2m}B \\<br />
\vdots & & \ddots & \\<br />
a_{n1}B & a_{n2}B & \cdots & a_{nm}B \end{array}\right). <br />
</math>|C.36}}<br />
<br />
Let us now consider two examples. First let <math>\left\vert\phi\right\rangle\,\!</math> and<br />
<math>\left\vert\psi\right\rangle\,\!</math> be as before,<br />
<center><math><br />
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right), \;\; <br />
\mbox{and} \;\; <br />
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{c} \gamma \\ \delta<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{c} \alpha\gamma\\ <br />
\alpha\delta \\<br />
\beta\gamma \\ <br />
\beta\delta <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.37}}<br />
Also<br />
{{Equation|<math>\begin{align}<br />
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\<br />
&= \left(\begin{array}{c} \alpha \\ \beta<br />
\end{array}\right) <br />
\otimes <br />
\left(\begin{array}{cc} \gamma^* & \delta^*<br />
\end{array}\right) \\<br />
&= \left(\begin{array}{cc}<br />
\alpha\gamma^* & \alpha\delta^* \\<br />
\beta\gamma^* & \beta\delta^* <br />
\end{array}\right). <br />
\end{align}<br />
</math>|C.38}}<br />
<br />
Now consider two matrices<br />
<center><math><br />
A = \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\;\; \mbox{and} \;\;<br />
B = \left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right). <br />
\,\!</math></center><br />
Then <br />
{{Equation|<math>\begin{align}<br />
A\otimes B &= \left(\begin{array}{cc} <br />
a & b \\<br />
c & d \end{array}\right), <br />
\otimes<br />
\left(\begin{array}{cc} <br />
e & f \\<br />
g & h \end{array}\right) \\ <br />
&= \left(\begin{array}{cccc} <br />
ae & af & be & bf \\<br />
ag & ah & bg & bh \\<br />
ce & cf & de & df \\<br />
cg & ch & dg & dh \end{array}\right). <br />
\end{align}<br />
</math>|C.39}}<br />
<br />
====Properties of Tensor Products====<br />
<br />
Some properties of tensor products which are useful are the following<br />
(with <math>A\,\!</math>, <math>B\,\!</math>, <math>C\,\!</math>, <math>D\,\!</math> any type):<br />
#<math>(A\otimes B)(C\otimes D) = AC \otimes BD\,\!</math><br />
#<math>(A\otimes B)^T = A^T\otimes B^T\,\!</math><br />
#<math>(A\otimes B)^* = A^*\otimes B^*\,\!</math><br />
#<math>(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!</math><br />
#<math>(A+B) \otimes C = A\otimes C+B\otimes C\,\!</math><br />
#<math>A\otimes(B+C) = A\otimes B + A\otimes C\,\!</math><br />
#<math>\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!</math><br />
(See Ref.~(\cite{Horn/JohnsonII}), Chapter 4.)</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Index&diff=813Talk:Index2010-04-12T16:36:52Z<p>Rceballos: </p>
<hr />
<div>I'm not finding a term in the text for "bra," "partial trace operator," "maximally mixed state, two qubits," "mean," "projection operator," "scalability," "X-gate," or "Y-gate." Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=812Index2010-04-12T16:35:06Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=811Index2010-04-12T16:33:09Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=810Index2010-04-12T16:31:31Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation [[Appendix A - Basic Probability Concepts|'''A''']]<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=809Index2010-04-12T16:29:32Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]]<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Index&diff=808Talk:Index2010-04-12T16:23:49Z<p>Rceballos: </p>
<hr />
<div>I'm not finding a term in the text for "bra," "partial trace operator," "maximally mixed state, two qubits," "mean," "projection operator," or "scalability." Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Index&diff=807Talk:Index2010-04-12T16:22:53Z<p>Rceballos: </p>
<hr />
<div>I'm not finding a term in the text for "bra," "partial trace operator," "maximally mixed state, two qubits," "mean," or "scalability." Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=806Index2010-04-12T16:22:06Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Index&diff=805Talk:Index2010-04-12T16:07:08Z<p>Rceballos: </p>
<hr />
<div>I'm not finding a term in the text for "bra," "partial trace operator," "maximally mixed state, two qubits," or "scalability." Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=804Index2010-04-12T16:05:26Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=803Index2010-04-12T16:03:33Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], <br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=802Index2010-04-12T01:28:45Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=801Index2010-04-12T01:27:35Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']], [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.3''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=800Index2010-04-12T01:24:52Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [[Chapter 2 - Qubits and Collections of Qubits#Standard Prescription|2.7.1]], [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']]<br />
:complex number [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Index&diff=799Talk:Index2010-04-12T01:18:24Z<p>Rceballos: </p>
<hr />
<div>I'm not finding a term in the text for "bra," "partial trace operator," or "scalability." Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Talk:Index&diff=798Talk:Index2010-04-12T01:18:10Z<p>Rceballos: </p>
<hr />
<div>I'm not finding a term in the text for "bracket," "partial trace operator," or "scalability." Russell</div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=797Index2010-04-12T01:17:17Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket [[Appendix A - Basic Probability Concepts#Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [['''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']]<br />
:complex number [['''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=796Index2010-04-12T01:15:13Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [['''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']]<br />
:complex number [['''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Chapter 2 - Qubits and Collections of Qubits#States of Many Qubits|2.5]], [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=795Index2010-04-12T01:12:46Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [['''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']]<br />
:complex number [['''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=794Index2010-04-12T01:09:53Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [['''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']]<br />
:complex number [['''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#No Cloning!|5.2]]<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballoshttps://www2.physics.siu.edu/qunet/wiki/index.php?title=Index&diff=790Index2010-04-11T00:32:35Z<p>Rceballos: </p>
<hr />
<div><div style="float: left; width: 32%"><br />
<br />
;<div id="A"><big>A</big></div><br />
:average - [[Appendix A - Basic Probability Concepts|'''A''']]<br />
<br />
;<div id="B"><big>B</big></div><br />
:basis vectors real [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
:bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:bit-flip operation [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Bloch Sphere [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:bra<br />
:bracket<br />
<br />
;<div id="C"><big>C</big></div><br />
:closed-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:CNOT gate(see controlled NOT) <br />
:commutator [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]], <br />
:complex conjugate [['''B''']]<br />
::of a matrix [[Appendix C - Vectors and Linear Algebra#Complex Conjugate|'''C.3.1''']]<br />
:complex number [['''B''']]<br />
:computational basis [[Chapter 2 - Qubits and Collections of Qubits#Qubit States|2.2]]<br />
:controlled NOT [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]], [[Chapter 2 - Qubits and Collections of Qubits#Many-qubit Circuits|2.6.2]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:controlled phase gate [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|6.1]]<br />
:controlled unitary operation [[Chapter 2 - Qubits and Collections of Qubits#Controlled Operations|2.6.1]]<br />
<br />
;<div id="D"><big>D</big></div><br />
:decoherence [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:degenerate [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:delta<br />
::Kronecker [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:dense coding [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]]<br />
:density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]],[[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::for two qubits [[Chapter 3 - Physics of Quantum Information#Density Matrix for a Mixed State: Two States|3.5.2]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
::mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
::pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:density operator [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:determinant [[Appendix C - Vectors and Linear Algebra#The Determinant|'''C.3.6''']]<br />
:DiVencenzo's requirements [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
:dot product<br />
<br />
;<div id="E"><big>E</big></div><br />
:eigenvalue decomposition [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:eigenvectors [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:entangled states (see entanglement)<br />
:entanglement [[Chapter 4 - Entanglement|4]], [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Quantum Dense Coding|5.4]], [[Chapter 1#How do quantum computers provide an advantage?|1.2.5]]<br />
::pure state [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
::mixed state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:expectation value [[Chapter 3 - Physics of Quantum Information#Expectation Values|3.6]]<br />
<br />
;<big>F</big><br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 32%"><br />
<br />
;<div id="G"><big>G</big></div><br />
:group [[Appendix D - Group Theory#Definitions and Examples|'''D.2''']]<br />
<br />
;<div id="H"><big>H</big></div><br />
:Hadamard gate [[Chapter 2 - Qubits and Collections of Qubits#eq2.16|2.16]]<br />
:Hamiltonian [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:Hermitian matrix [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5]], [[Chapter 8 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation|8.2]], [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]], [[Appendix C - Vectors and Linear Algebra#Hermitian Conjugate|'''C.3.3''']], [[Appendix C - Vectors and Linear Algebra#Examples|'''C.6.1''']], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
:Hilbert-Schmidt inner product [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]]<br />
<br />
;<div id="I"><big>I</big></div><br />
:inner product <br />
::for real vectors [[Appendix C - Vectors and Linear Algebra#Real Vectors|'''C.2.1''']]<br />
::for complex vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:inverse of a matrix [[Appendix C - Vectors and Linear Algebra#The Inverse of a Matrix|'''C.3.7''']]<br />
<br />
;<div id="K"><big>K</big></div><br />
:ket [[Appendix C - Vectors and Linear Algebra#Complex Vectors|'''C.2.2''']]<br />
:Kraus operators [[Chapter 8 - Noise in Quantum Systems#Physics Behind the Noise and Completely Positive Maps|8.3]]<br />
:Kronecker delta [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']]<br />
:Kronecker product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
<br />
;<div id="L"><big>L</big></div><br />
:local operations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]]<br />
:local unitary transformations [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Chapter 4 - Entanglement#Bell States|4.2.1]]<br />
<br />
;<div id="M"><big>M</big></div><br />
:matrix exponentiation [[Chapter 3 - Physics of Quantum Information#expmatrix|3.2]]<br />
:maximally entangled states [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:maximally mixed state [[Chapter 3 - Physics of Quantum Information#Density Matrix for the Description of Open Quantum Systems: An Example|3.5.3]]<br />
::two qubits<br />
:mean<br />
:median<br />
:mixed state density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Mixed States|3.5]]<br />
:modulus squared [[Appendix B - Complex Numbers#Appendix B - Complex Numbers|'''B''']]<br />
<br />
;<div id="O"><big>O</big></div><br />
:open quantum systems [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:open-system evolution [[Chapter 1#Obstacles to Building a Reliable Quantum Computer|1.4]]<br />
:operator-sum decomposition [[Chapter 8 - Noise in Quantum Systems#Unitary Degree of Freedom in the OSR|8.4]]<br />
:orthogonal [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[<br />
::vectors [[Appendix C - Vectors and Linear Algebra#More Dirac Notation|'''C.4''']], [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
<br />
;<div id="P"><big>P</big></div><br />
:partial trace<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:Pauli matrices [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|2.4]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|3.4]], [[Chapter 3 - Physics of Quantum Information#Two-State Example: Bloch Sphere|3.5.4]]<br />
:phase gate [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:phase-flip [[Chapter 2 - Qubits and Collections of Qubits#Examples of Important Qubit Gates|2.3.2]]<br />
:Planck's constant [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.2]]<br />
:projection operator<br />
:pure state [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]], [[Chapter 4 - Entanglement#Entangled Pure States|4.2]], [[Appendix E - Density Operator: Extensions#Appendix E - Density Operator: Extensions|'''E''']]<br />
<br />
<br />
</div><div style="float: left; width: 2%"><br />
</div><div style="float: left; width: 31%"><br />
<br />
;<div id="Q"><big>Q</big></div><br />
:Qbit (see qubit)<br />
:quantum bit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
:quantum dense coding (see [[#D|dense coding]])<br />
:quantum gates [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]], [[Chapter 2 - Qubits and Collections of Qubits#Qubit Gates|2.3]], [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:qubit [[Chapter 1#Bits and qubits: An Introduction|1.3]]<br />
<br />
;<div id="R"><big>R</big></div><br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::of a Bell state [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:reduced density matrix [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
::see reduced density operator<br />
:reduced density operator [[Chapter 4 - Entanglement#Reduced Density Operators and the Partial Trace|4.3.1]]<br />
:requirements for scalable quantum computing [[Chapter 2 - Qubits and Collections of Qubits#Introduction|2.1]]<br />
<br />
;<div id="S"><big>S</big></div><br />
:scalability<br />
:Schrodinger Equation [[Chapter 3 - Physics of Quantum Information#Schrodinger's Equation|3.1]]<br />
::for density matrix [[Chapter 3 - Physics of Quantum Information#Density Matrix for Pure States|3.3]]<br />
:separable state [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
::simply separable [[Chapter 4 - Entanglement#Entangled Mixed States|4.3]]<br />
:similar matrices [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:similarity transformation [[Appendix C - Vectors and Linear Algebra#Transformations|'''C.5''']]<br />
:singular values [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:special unitary matrix<br />
:spectrum [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']]<br />
:standard deviation<br />
:SU [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']]<br />
<br />
;<div id="T"><big>T</big></div><br />
:teleportation [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Teleporting a Quantum State|5.5]]<br />
:tensor product [[Appendix C - Vectors and Linear Algebra#Tensor Products|'''C.7''']]<br />
:trace [[Appendix C - Vectors and Linear Algebra#The Trace|'''C.3.5''']]<br />
::partial(see partial trace)<br />
:transpose [[Appendix C - Vectors and Linear Algebra#Transpose|'''C.3.2''']]<br />
<br />
;<div id="U"><big>U</big></div><br />
:uncertainty principle [[Chapter 5 - Quantum Information: Basic Principles and Simple Examples#Uncertainty Principle|5.3]]<br />
:unitary matrix [[Chapter 2 - Qubits and Collections of Qubits#Chapter 2 - Qubits and Collections of Qubits|2.3]], [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|'''C.3.8''']], [[Appendix D - Group Theory#Infinite Order Groups: Lie Groups|'''D.7.2''']]<br />
:universal set of gates [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
:universality [[Chapter 2 - Qubits and Collections of Qubits#Quantum Gates for Many Qubits|2.6]]<br />
<br />
;<div id="V"><big>V</big></div><br />
:variance<br />
<br />
;<div id="X"><big>X</big></div><br />
:X-gate<br />
<br />
;<div id="Y"><big>Y</big></div><br />
:Y-gate<br />
<br />
;<div id="Z"><big>Z</big></div><br />
:Z-gate<br />
<br />
<br />
<br />
</div></div>Rceballos