<?xml version="1.0"?>
<feed xmlns="http://www.w3.org/2005/Atom" xml:lang="en">
	<id>https://www2.physics.siu.edu/qunet/wiki/api.php?action=feedcontributions&amp;feedformat=atom&amp;user=Hilary</id>
	<title>Qunet - User contributions [en]</title>
	<link rel="self" type="application/atom+xml" href="https://www2.physics.siu.edu/qunet/wiki/api.php?action=feedcontributions&amp;feedformat=atom&amp;user=Hilary"/>
	<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php/Special:Contributions/Hilary"/>
	<updated>2026-04-10T02:12:05Z</updated>
	<subtitle>User contributions</subtitle>
	<generator>MediaWiki 1.31.7</generator>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2372</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2372"/>
		<updated>2013-10-18T04:56:36Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction Hamiltonian:''' When this Hamiltonian is between two qubits labelled &amp;lt;math&amp;gt; i \,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; j \,\!&amp;lt;/math&amp;gt;, it can be expressed as &amp;lt;math&amp;gt; &lt;br /&gt;
H_{ex}^{i,j} =  X_iX_j + Y_iY_j + Z_iZ_j,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt; X_i \,\!&amp;lt;/math&amp;gt; is the Pauli x-operation on the &amp;lt;math&amp;gt; i\,\!&amp;lt;/math&amp;gt;th qubit and similarly for the other operators.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' Product of two Hilbert-Schmidt operators.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt operators:''' Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:''' Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = \mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:''' Property of certain binary operations that are possessed by Lie groups that can be written &amp;lt;math&amp;gt;&lt;br /&gt;
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:''' The symbol &amp;lt;math&amp;gt;\delta_{ij}\!&amp;lt;/math&amp;gt; that is defined as:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\delta_{ij} = \begin{cases}&lt;br /&gt;
               1, &amp;amp; \mbox{if } i=j, \\&lt;br /&gt;
               0, &amp;amp; \mbox{if } i\neq j.&lt;br /&gt;
              \end{cases}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:''' A vector space, over some field with a binary operation, called a Lie bracket, that possesses bilinearity, is alternating on the vector space, and satisfies the Jacobi identity.&lt;br /&gt;
&lt;br /&gt;
'''Lie group:''' A differentiable manifold that corresponds to a continuous set of symmetries.&lt;br /&gt;
&lt;br /&gt;
'''Linear code:''' Error correcting code for which any linear combination of codewords is a codeword. Can be generated by a Generator matrix.&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:''' A very general mapping of a matrix acting on a vector that produces another vector. &lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:''' See ''Local operations''.&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Local operations:'''  Actions on an individual particle without involving any other particle.&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:''' Short for ''Logical qubit''. Qubits that are encoded with information that is to be protected from errors. Represented by &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; codes.&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:''' A mapping of the function of a matrix according to its Taylor expansion.&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:''' Defining characteristics that allow matrices to be categorized.&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:''' Representation of a change in basis from one matrix to another.&lt;br /&gt;
&lt;br /&gt;
'''Measurement:''' In quantum mechanics, a probability distribution that determines the state of the quantum object. Currently, it is impossible to measure a quantum object without affecting or destroying its current state.&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:''' The smallest Hamming distance between any two non-equal vectors in a code.&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:''' Complex norm of the expression &amp;lt;math&amp;gt;\alpha + \beta i\!&amp;lt;/math&amp;gt; that can be expressed by &amp;lt;math&amp;gt;|\alpha + \beta i| = |z| = \sqrt{\alpha^2 + \beta^2}, \alpha,\beta\in \mathbb{R}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:''' Errors in a system created by unwanted interactions with other systems or imperfect controls.&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:''' Codes that have eigenvalues with multiplicity of exactly one.&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:''' A subgroup &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; of a group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt; that leaves the set &amp;lt;math&amp;gt;\mathcal{M}\,\!&amp;lt;/math&amp;gt; fixed and can be expressed by &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element. Injective.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element. Surjective.&lt;br /&gt;
&lt;br /&gt;
'''Open system:''' A system that is not necessarily conserved as it is exposed to an external environment. An environment in which noise may be created.&lt;br /&gt;
&lt;br /&gt;
'''Operator:''' The basis of theory in quantum mechanics. A function which acts on information obtained in an environment to describe the environment and some relevant characteristics of that data.&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:''' The number of elements contained in a group.&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:''' An organizational method that lists the basis of a vector in a logical order.&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:''' Tensor product of two vectors.&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:''' The property of an equation that either remains the same (''even parity'') or changes (''odd parity'').&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:''' Matrix that can be found using the generator matrix. It has the property that it annihilates a code word.&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''  The trace over one of the subsystems (particle states) of a composite system.&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:''' A division of a group into disjoint, nonempty sets.&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:''' The three matrices that, along with the &amp;lt;math&amp;gt;2 \times 2\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
identity matrix, form a basis for the set of &amp;lt;math&amp;gt;2 \times 2\,\!&amp;lt;/math&amp;gt; Hermitian matrices and can be used to&lt;br /&gt;
describe all &amp;lt;math&amp;gt;2 \times 2&amp;lt;/math&amp;gt; unitary transformations. &lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The &amp;lt;math&amp;gt;X,Y,Z\!&amp;lt;/math&amp;gt; gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:''' A bijective map from a set to itself.&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:''' In this case a &amp;lt;math&amp;gt;\left\vert 1\right\rangle\,\!&amp;lt;/math&amp;gt; will acquire a (-1) sign change due to some noise, but a  &amp;lt;math&amp;gt;\left\vert 0\right\rangle\,\!&amp;lt;/math&amp;gt; is unaffected.  If a quantum phase-flip error occurs with some probability &amp;lt;math&amp;gt;p\,\!&amp;lt;/math&amp;gt;, we may express the phase flip error as &amp;lt;math&amp;gt;&lt;br /&gt;
\rho^\prime = (1-p)\rho + p \sigma_z \rho \sigma_z&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See ''Z gate''.&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''  A constant that is a quantum of action in quantum mechanics. The value of Planck's constant is &amp;lt;math&amp;gt;h = 6.626 \times 10^{-34} J\cdot s \!&amp;lt;/math&amp;gt;. &lt;br /&gt;
&lt;br /&gt;
'''Polarization:''' The process that forces waves to only oscillate in one plane.&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:''' The probability that a quantum system is in the state &amp;lt;math&amp;gt;\psi_n\!&amp;lt;/math&amp;gt;. It can be expressed as &amp;lt;math&amp;gt;\left\vert\psi_n\right\rangle = \alpha_0\left\vert 0\right\rangle +&lt;br /&gt;
    \alpha_1\left\vert 1\right\rangle,&lt;br /&gt;
\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;|\alpha_0|^2 + |\alpha_1|^2 = 1\,\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;|\alpha_0|^2\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;|\alpha_1|^2\!&amp;lt;/math&amp;gt; are the probabilities.&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:''' If we measure a system, i.e. look to see if it is Well 0 or Well 1, we will “project it into&lt;br /&gt;
one state or the other.” In other words, suppose the system is in the state &amp;lt;math&amp;gt;\left|\psi\right\rangle\,\!&amp;lt;/math&amp;gt; above. If we&lt;br /&gt;
look to see where the particle is and find it in Well 1, then the probability is clearly zero &lt;br /&gt;
that it is in the other well.&lt;br /&gt;
&lt;br /&gt;
'''Pure state:''' A state that cannot be expressed as a mixture of other states.&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:''' Use of quantum mechanics to encode messages.&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:''' Sending two bits of information only using one qubit, using entanglement.&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:''' See ''Hamming Bound''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:''' Use of quantum cryptography to generate a shared random key.&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:''' Describes the number of states that make up a density operator. A rank one projection is a pure state density operator.&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:''' Given by the ration of the number of logical bits to the number of bits, &amp;lt;math&amp;gt;k/n\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:''' A test for observable quantum characteristics that uses the partial trace operation.&lt;br /&gt;
&lt;br /&gt;
'''Representation space:''' A vector space used to describe quantum fields.&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:''' A public-key cryptography algorithm which uses prime factorization and is rendered useless by Shor's algorithm.&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:''' A method that takes a density operator and transforms it into a reduced density operator.&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:''' Partial differential equation that describes how the quantum states of physical systems change with time.&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:''' Used to find prime factors, named after its inventor, Peter Shor.&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''  The first example of a quantum error correcting code which, in principle, can correct arbitrary single-qubit errors and can be understood in terms of the simple classical error correcting code.&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:''' A matrix transformation that leave both the trace and the determinant of a matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Singular values:''' The entries of the diagonal matrix, &amp;lt;math&amp;gt;D\!&amp;lt;/math&amp;gt;, that are produced from some complex matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:''' A form in which a complex matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; has a diagonal matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; such that &lt;br /&gt;
&amp;lt;math&amp;gt; M = UDV,  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;V\,\!&amp;lt;/math&amp;gt; are unitary matrices.&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:''' Method for representing open system evolution. [[Chapter 6 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation |Full description here.]]&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:''' The set of special unitary matrices is denoted &amp;lt;math&amp;gt;SU(n)\,\!&amp;lt;/math&amp;gt; and has determinant of one.&lt;br /&gt;
&lt;br /&gt;
'''Spin:''' A two-state system being either up or down when measured.&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:''' Used to describe entangled states at great distances. For example, an entangled electron-positron pair are at opposite ends of the galaxy when one of them is measured. The conservation of angular momentum says that the other particle all of the way on the other side of the galaxy must instantly be the opposite spin as the measured particle. An idea explored in the EPR paradox.&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:''' The subgroup &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; of a group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt; that leaves the element &amp;lt;math&amp;gt;m\,\!&amp;lt;/math&amp;gt; fixed: &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{S} = \{S\in \mathcal{S}|sm=m\}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:''' Family of quantum error correcting codes which are describable by using the stabilizer of a state (really a set of states) in the Hilbert space, conveniently described by their operators rather than their states.&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:''' The amount of variation from the mean.&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:''' A formula used to approximate &amp;lt;math&amp;gt;n!\!&amp;lt;/math&amp;gt;, denoted by &amp;lt;math&amp;gt; n! ~= \sqrt{2\pi n}(\frac{n}{e})^n.&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:''' A subgroup &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; of a group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt; is a subset of the group elements that satisfies all&lt;br /&gt;
the properties in the definition of a group under the inherited multiplication rule.&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:''' Measurement to find the error syndrome that gives information about the error and not the actual code word.&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:''' can be used to exponentiate a matrix by letting the matrix replace &amp;lt;math&amp;gt;x\!&amp;lt;/math&amp;gt; in the equation, &amp;lt;math&amp;gt;&lt;br /&gt;
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}&lt;br /&gt;
.\!&amp;lt;/math&amp;gt;&lt;br /&gt;
'''Teleportation:''' process by which quantum information can be transmitted from one location to another using entanglement.&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''  Used extensively in quantum mechanics. It is commonly denoted with a &amp;lt;math&amp;gt;\otimes\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
symbol, although this is often left out.&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''  The same set of elements, but now the first row becomes the first column, the second row becomes the second column, and so on. Thus the rows and columns are interchanged.&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:''' Map of a group where all of the elements of the group act as identity operators of the map.&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:''' Machine used to emulate classic computational algorithms. &lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:''' Inequalities that describe the limitations of the measurements of quantum objects that can be made simultaneously.&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''  One whose&lt;br /&gt;
inverse is also its Hermitian conjugate, &amp;lt;math&amp;gt;U^\dagger = U^{-1}\,\!&amp;lt;/math&amp;gt;, so that &amp;lt;math&amp;gt;U^\dagger U = UU^\dagger = \mathbb{I}&lt;br /&gt;
.\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:''' The ability to implement a special class of two-qubit gates between any two qubits, plus the ability to implement all single-qubit unitary transformations.&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality, or the ability to perform any arbitrary computation with quantum computing.&lt;br /&gt;
&lt;br /&gt;
'''Variance:''' How far a set of numbers is spread out from the mean.&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A quantity with both magnitude and direction.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:''' Collection of vectors, added together and multiplied by scalars, used to describe a space that is the focus of studies in linear algebra.&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:''' Coefficients that allow a new basis to be expressed in terms of the old basis. They also block-diagonalize the tensor products of the operators.&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' A bit-flip gate, represented by the matrix &amp;lt;math&amp;gt;X = \left(\begin{array}{cc} 0 &amp;amp; 1 \\ &lt;br /&gt;
                      1 &amp;amp; 0 \end{array}\right).&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Y gate:''' Acts on states with both &amp;lt;math&amp;gt;X\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;Z\!&amp;lt;/math&amp;gt; gates, represented by the matrix &amp;lt;math&amp;gt;Y =  \left(\begin{array}{cc} 0 &amp;amp; -i \\ &lt;br /&gt;
                      i &amp;amp; 0 \end{array}\right).&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate, represented by the matrix &amp;lt;math&amp;gt;Z = \left(\begin{array}{cc} 1 &amp;amp; 0 \\ 0 &amp;amp; -1 \end{array}\right).&amp;lt;/math&amp;gt;&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2371</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2371"/>
		<updated>2013-10-18T03:53:13Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction Hamiltonian:''' When this Hamiltonian is between two qubits labelled &amp;lt;math&amp;gt; i \,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; j \,\!&amp;lt;/math&amp;gt;, it can be expressed as &amp;lt;math&amp;gt; &lt;br /&gt;
H_{ex}^{i,j} =  X_iX_j + Y_iY_j + Z_iZ_j,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt; X_i \,\!&amp;lt;/math&amp;gt; is the Pauli x-operation on the &amp;lt;math&amp;gt; i\,\!&amp;lt;/math&amp;gt;th qubit and similarly for the other operators.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' Product of two Hilbert-Schmidt operators.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt operators:''' Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:''' Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = \mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:''' Property of certain binary operations that are possessed by Lie groups that can be written &amp;lt;math&amp;gt;&lt;br /&gt;
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:''' The symbol &amp;lt;math&amp;gt;\delta_{ij}\!&amp;lt;/math&amp;gt; that is defined as:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\delta_{ij} = \begin{cases}&lt;br /&gt;
               1, &amp;amp; \mbox{if } i=j, \\&lt;br /&gt;
               0, &amp;amp; \mbox{if } i\neq j.&lt;br /&gt;
              \end{cases}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:''' A vector space, over some field with a binary operation, called a Lie bracket, that possesses bilinearity, is alternating on the vector space, and satisfies the Jacobi identity.&lt;br /&gt;
&lt;br /&gt;
'''Lie group:''' A differentiable manifold that corresponds to a continuous set of symmetries.&lt;br /&gt;
&lt;br /&gt;
'''Linear code:''' Error correcting code for which any linear combination of codewords is a codeword. Can be generated by a Generator matrix.&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:''' A very general mapping of a matrix acting on a vector that produces another vector. &lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:''' See ''Local operations''.&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Local operations:'''  Actions on an individual particle without involving any other particle.&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:''' Short for ''Logical qubit''. Qubits that are encoded with information that is to be protected from errors. Represented by &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; codes.&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:''' A mapping of the function of a matrix according to its Taylor expansion.&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:''' Defining characteristics that allow matrices to be categorized.&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:''' Representation of a change in basis from one matrix to another.&lt;br /&gt;
&lt;br /&gt;
'''Measurement:''' In quantum mechanics, a probability distribution that determines the state of the quantum object. Currently, it is impossible to measure a quantum object without affecting or destroying its current state.&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:''' The smallest Hamming distance between any two non-equal vectors in a code.&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:''' Complex norm of the expression &amp;lt;math&amp;gt;\alpha + \beta i\!&amp;lt;/math&amp;gt; that can be expressed by &amp;lt;math&amp;gt;|\alpha + \beta i| = |z| = \sqrt{\alpha^2 + \beta^2}, \alpha,\beta\in \mathbb{R}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:''' Errors in a system created by unwanted interactions with other systems or imperfect controls.&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:''' Codes that have eigenvalues with multiplicity of exactly one.&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:''' A subgroup &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; of a group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt; that leaves the set &amp;lt;math&amp;gt;\mathcal{M}\,\!&amp;lt;/math&amp;gt; fixed and can be expressed by &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element. Injective.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element. Surjective.&lt;br /&gt;
&lt;br /&gt;
'''Open system:''' A system that is not necessarily conserved as it is exposed to an external environment. An environment in which noise may be created.&lt;br /&gt;
&lt;br /&gt;
'''Operator:''' The basis of theory in quantum mechanics. A function which acts on information obtained in an environment to describe the environment and some relevant characteristics of that data.&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:''' The number of elements contained in a group.&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:''' An organizational method that lists the basis of a vector in a logical order.&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:''' Tensor product of two vectors.&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:''' The property of an equation that either remains the same (''even parity'') or changes (''odd parity'').&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:''' Matrix that can be found using the generator matrix. It has the property that it annihilates a code word.&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''  The trace over one of the subsystems (particle states) of a composite system.&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:''' A division of a group into disjoint, nonempty sets.&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:''' The three matrices that, along with the &amp;lt;math&amp;gt;2 \times 2\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
identity matrix, form a basis for the set of &amp;lt;math&amp;gt;2 \times 2\,\!&amp;lt;/math&amp;gt; Hermitian matrices and can be used to&lt;br /&gt;
describe all &amp;lt;math&amp;gt;2 \times 2&amp;lt;/math&amp;gt; unitary transformations. &lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The &amp;lt;math&amp;gt;X,Y,Z\!&amp;lt;/math&amp;gt; gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:''' A bijective map from a set to itself.&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:''' In this case a &amp;lt;math&amp;gt;\left\vert 1\right\rangle\,\!&amp;lt;/math&amp;gt; will acquire a (-1) sign change due to some noise, but a  &amp;lt;math&amp;gt;\left\vert 0\right\rangle\,\!&amp;lt;/math&amp;gt; is unaffected.  If a quantum phase-flip error occurs with some probability &amp;lt;math&amp;gt;p\,\!&amp;lt;/math&amp;gt;, we may express the phase flip error as &amp;lt;math&amp;gt;&lt;br /&gt;
\rho^\prime = (1-p)\rho + p \sigma_z \rho \sigma_z&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See ''Z gate''.&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''  A constant that is a quantum of action in quantum mechanics. The value of Planck's constant is &amp;lt;math&amp;gt;h = 6.626 \times 10^{-34} J\cdot s \!&amp;lt;/math&amp;gt;. &lt;br /&gt;
&lt;br /&gt;
'''Polarization:''' The process that forces waves to only oscillate in one plane.&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:''' The probability that a quantum system is in the state &amp;lt;math&amp;gt;\psi_n\!&amp;lt;/math&amp;gt;. It can be expressed as &amp;lt;math&amp;gt;\left\vert\psi_n\right\rangle = \alpha_0\left\vert 0\right\rangle +&lt;br /&gt;
    \alpha_1\left\vert 1\right\rangle,&lt;br /&gt;
\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;|\alpha_0|^2 + |\alpha_1|^2 = 1\,\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;|\alpha_0|^2\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;|\alpha_1|^2\!&amp;lt;/math&amp;gt; are the probabilities.&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:''' If we measure a system, i.e. look to see if it is Well 0 or Well 1, we will “project it into&lt;br /&gt;
one state or the other.” In other words, suppose the system is in the state &amp;lt;math&amp;gt;\left|\psi\right\rangle\,\!&amp;lt;/math&amp;gt; above. If we&lt;br /&gt;
look to see where the particle is and find it in Well 1, then the probability is clearly zero &lt;br /&gt;
that it is in the other well.&lt;br /&gt;
&lt;br /&gt;
'''Pure state:''' A state that cannot be expressed as a mixture of other states.&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:''' Use of quantum mechanics to encode messages.&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:''' Sending two bits of information only using one qubit, using entanglement.&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:''' See ''Hamming Bound''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:''' Use of quantum cryptography to generate a shared random key.&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:''' Describes the number of states that make up a density operator. A rank one projection is a pure state density operator.&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:''' Given by the ration of the number of logical bits to the number of bits, &amp;lt;math&amp;gt;k/n\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:''' A test for observable quantum characteristics that uses the partial trace operation.&lt;br /&gt;
&lt;br /&gt;
'''Representation space:''' A vector space used to describe quantum fields.&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:''' A public-key cryptography algorithm which uses prime factorization and is rendered useless by Shor's algorithm.&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:''' A method that takes a density operator and transforms it into a reduced density operator.&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:''' Partial differential equation that describes how the quantum states of physical systems change with time.&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:''' Used to find prime factors, named after its inventor, Peter Shor.&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''  The first example of a quantum error correcting code which, in principle, can correct arbitrary single-qubit errors and can be understood in terms of the simple classical error correcting code.&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:''' A matrix transformation that leave both the trace and the determinant of a matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Singular values:''' The entries of the diagonal matrix, &amp;lt;math&amp;gt;D\!&amp;lt;/math&amp;gt;, that are produced from some complex matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:''' A form in which a complex matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; has a diagonal matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; such that &lt;br /&gt;
&amp;lt;math&amp;gt; M = UDV,  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;V\,\!&amp;lt;/math&amp;gt; are unitary matrices.&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:''' Method for representing open system evolution. [[Chapter 6 - Noise in Quantum Systems#SMR Representation or Operator-Sum Representation |Full description here.]]&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:''' The set of special unitary matrices is denoted &amp;lt;math&amp;gt;SU(n)\,\!&amp;lt;/math&amp;gt; and has determinant of one.&lt;br /&gt;
&lt;br /&gt;
'''Spin:''' A two-state system being either up or down when measured.&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2370</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2370"/>
		<updated>2013-10-18T02:29:11Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction Hamiltonian:''' When this Hamiltonian is between two qubits labelled &amp;lt;math&amp;gt; i \,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; j \,\!&amp;lt;/math&amp;gt;, it can be expressed as &amp;lt;math&amp;gt; &lt;br /&gt;
H_{ex}^{i,j} =  X_iX_j + Y_iY_j + Z_iZ_j,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt; X_i \,\!&amp;lt;/math&amp;gt; is the Pauli x-operation on the &amp;lt;math&amp;gt; i\,\!&amp;lt;/math&amp;gt;th qubit and similarly for the other operators.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' Product of two Hilbert-Schmidt operators.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt operators:''' Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:''' Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = \mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:''' Property of certain binary operations that are possessed by Lie groups that can be written &amp;lt;math&amp;gt;&lt;br /&gt;
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:''' The symbol &amp;lt;math&amp;gt;\delta_{ij}\!&amp;lt;/math&amp;gt; that is defined as:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\delta_{ij} = \begin{cases}&lt;br /&gt;
               1, &amp;amp; \mbox{if } i=j, \\&lt;br /&gt;
               0, &amp;amp; \mbox{if } i\neq j.&lt;br /&gt;
              \end{cases}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:''' A vector space, over some field with a binary operation, called a Lie bracket, that possesses bilinearity, is alternating on the vector space, and satisfies the Jacobi identity.&lt;br /&gt;
&lt;br /&gt;
'''Lie group:''' A differentiable manifold that corresponds to a continuous set of symmetries.&lt;br /&gt;
&lt;br /&gt;
'''Linear code:''' Error correcting code for which any linear combination of codewords is a codeword. Can be generated by a Generator matrix.&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:''' A very general mapping of a matrix acting on a vector that produces another vector. &lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:''' See ''Local operations''.&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Local operations:'''  Actions on an individual particle without involving any other particle.&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:''' Short for ''Logical qubit''. Qubits that are encoded with information that is to be protected from errors. Represented by &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; codes.&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:''' A mapping of the function of a matrix according to its Taylor expansion.&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:''' Defining characteristics that allow matrices to be categorized.&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:''' Representation of a change in basis from one matrix to another.&lt;br /&gt;
&lt;br /&gt;
'''Measurement:''' In quantum mechanics, a probability distribution that determines the state of the quantum object. Currently, it is impossible to measure a quantum object without affecting or destroying its current state.&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:''' The smallest Hamming distance between any two non-equal vectors in a code.&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:''' Complex norm of the expression &amp;lt;math&amp;gt;\alpha + \beta i\!&amp;lt;/math&amp;gt; that can be expressed by &amp;lt;math&amp;gt;|\alpha + \beta i| = |z| = \sqrt{\alpha^2 + \beta^2}, \alpha,\beta\in \mathbb{R}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:''' Errors in a system created by unwanted interactions with other systems or imperfect controls.&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:''' Codes that have eigenvalues with multiplicity of exactly one.&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:''' A subgroup &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; of a group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt; that leaves the set &amp;lt;math&amp;gt;\mathcal{M}\,\!&amp;lt;/math&amp;gt; fixed and can be expressed by &amp;lt;math&amp;gt;&lt;br /&gt;
\mathcal{S} = \{S\in \mathcal{S}|S\mathcal{M}=\mathcal{M}\}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element. Injective.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element. Surjective.&lt;br /&gt;
&lt;br /&gt;
'''Open system:''' A system that is not necessarily conserved as it is exposed to an external environment. An environment in which noise may be created.&lt;br /&gt;
&lt;br /&gt;
'''Operator:''' The basis of theory in quantum mechanics. A function which acts on information obtained in an environment to describe the environment and some relevant characteristics of that data.&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:''' The number of elements contained in a group.&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:''' An organizational method that lists the basis of a vector in a logical order.&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:''' Tensor product of two vectors.&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:''' The property of an equation that either remains the same (''even parity'') or changes (''odd parity'').&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:''' Matrix that can be found using the generator matrix. It has the property that it annihilates a code word.&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''  The trace over one of the subsystems (particle states) of a composite system.&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:''' A division of a group into disjoint, nonempty sets.&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:''' The three matrices that, along with the &amp;lt;math&amp;gt;2 \times 2\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
identity matrix, form a basis for the set of &amp;lt;math&amp;gt;2 \times 2\,\!&amp;lt;/math&amp;gt; Hermitian matrices and can be used to&lt;br /&gt;
describe all &amp;lt;math&amp;gt;2 \times 2&amp;lt;/math&amp;gt; unitary transformations. &lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The &amp;lt;math&amp;gt;X,Y,Z\!&amp;lt;/math&amp;gt; gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:''' A bijective map from a set to itself.&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:''' In this case a &amp;lt;math&amp;gt;\left\vert 1\right\rangle\,\!&amp;lt;/math&amp;gt; will acquire a (-1) sign change due to some noise, but a  &amp;lt;math&amp;gt;\left\vert 0\right\rangle\,\!&amp;lt;/math&amp;gt; is unaffected.  If a quantum phase-flip error occurs with some probability &amp;lt;math&amp;gt;p\,\!&amp;lt;/math&amp;gt;, we may express the phase flip error as &amp;lt;math&amp;gt;&lt;br /&gt;
\rho^\prime = (1-p)\rho + p \sigma_z \rho \sigma_z&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See ''Z gate''.&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''  A constant that is a quantum of action in quantum mechanics. The value of Planck's constant is &amp;lt;math&amp;gt;h = 6.626 \times 10^{-34} J\cdot s \!&amp;lt;/math&amp;gt;. &lt;br /&gt;
&lt;br /&gt;
'''Polarization:''' The process that forces waves to only oscillate in one plane.&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:''' The probability that a quantum system is in the state &amp;lt;math&amp;gt;\psi_n\!&amp;lt;/math&amp;gt;. It can be expressed as &amp;lt;math&amp;gt;\left\vert\psi_n\right\rangle = \alpha_0\left\vert 0\right\rangle +&lt;br /&gt;
    \alpha_1\left\vert 1\right\rangle,&lt;br /&gt;
\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;|\alpha_0|^2 + |\alpha_1|^2 = 1\,\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;|\alpha_0|^2\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;|\alpha_1|^2\!&amp;lt;/math&amp;gt; are the probabilities.&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:''' If we measure a system, i.e. look to see if it is Well 0 or Well 1, we will “project it into&lt;br /&gt;
one state or the other.” In other words, suppose the system is in the state &amp;lt;math&amp;gt;\left|\psi\right\rangle\,\!&amp;lt;/math&amp;gt; above. If we&lt;br /&gt;
look to see where the particle is and find it in Well 1, then the probability is clearly zero &lt;br /&gt;
that it is in the other well.&lt;br /&gt;
&lt;br /&gt;
'''Pure state:''' A state that cannot be expressed as a mixture of other states.&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:''' Use of quantum mechanics to encode messages.&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:''' Sending two bits of information only using one qubit, using entanglement.&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:''' See ''Hamming Bound''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:''' Use of quantum cryptography to generate a shared random key.&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:''' Describes the number of states that make up a density operator. A rank one projection is a pure state density operator.&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:''' Given by the ration of the number of logical bits to the number of bits, &amp;lt;math&amp;gt;k/n\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:''' &lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2369</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2369"/>
		<updated>2013-10-17T22:37:43Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction Hamiltonian:''' When this Hamiltonian is between two qubits labelled &amp;lt;math&amp;gt; i \,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; j \,\!&amp;lt;/math&amp;gt;, it can be expressed as &amp;lt;math&amp;gt; &lt;br /&gt;
H_{ex}^{i,j} =  X_iX_j + Y_iY_j + Z_iZ_j,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt; X_i \,\!&amp;lt;/math&amp;gt; is the Pauli x-operation on the &amp;lt;math&amp;gt; i\,\!&amp;lt;/math&amp;gt;th qubit and similarly for the other operators.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' Product of two Hilbert-Schmidt operators.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt operators:''' Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:''' Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:''' Property of certain binary operations that are possessed by Lie groups that can be written &amp;lt;math&amp;gt;&lt;br /&gt;
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:''' The symbol &amp;lt;math&amp;gt;\delta_{ij}\!&amp;lt;/math&amp;gt; that is defined as:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\delta_{ij} = \begin{cases}&lt;br /&gt;
               1, &amp;amp; \mbox{if } i=j, \\&lt;br /&gt;
               0, &amp;amp; \mbox{if } i\neq j.&lt;br /&gt;
              \end{cases}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:''' A vector space, over some field with a binary operation, called a Lie bracket, that possesses bilinearity, is alternating on the vector space, and satisfies the Jacobi identity.&lt;br /&gt;
&lt;br /&gt;
'''Lie group:''' A differentiable manifold that corresponds to a continuous set of symmetries.&lt;br /&gt;
&lt;br /&gt;
'''Linear code:''' Error correcting code for which any linear combination of codewords is a codeword. Can be generated by a Generator matrix.&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:''' A very general mapping of a matrix acting on a vector that produces another vector. &lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:''' See ''Local operations''.&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Local operations:'''  Actions on an individual particle without involving any other particle.&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:''' Short for ''Logical qubit''. Qubits that are encoded with information that is to be protected from errors. Represented by &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; codes.&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:''' A mapping of the function of a matrix according to its Taylor expansion.&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:''' Defining characteristics that allow matrices to be categorized.&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:''' Representation of a change in basis from one matrix to another.&lt;br /&gt;
&lt;br /&gt;
'''Measurement:''' In quantum mechanics, a probability distribution that determines the state of the quantum object. Currently, it is impossible to measure a quantum object without affecting or destroying its current state.&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:''' The smallest Hamming distance between any two non-equal vectors in a code.&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:''' Complex norm of the expression &amp;lt;math&amp;gt;\alpha + \beta i\!&amp;lt;/math&amp;gt; that can be expressed by &amp;lt;math&amp;gt;|\alpha + \beta i| = |z| = \sqrt{\alpha^2 + \beta^2}, \alpha,\beta\in \mathbb{R}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2368</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2368"/>
		<updated>2013-10-17T22:02:52Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction Hamiltonian:''' When this Hamiltonian is between two qubits labelled &amp;lt;math&amp;gt; i \,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; j \,\!&amp;lt;/math&amp;gt;, it can be expressed as &amp;lt;math&amp;gt; &lt;br /&gt;
H_{ex}^{i,j} =  X_iX_j + Y_iY_j + Z_iZ_j,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt; X_i \,\!&amp;lt;/math&amp;gt; is the Pauli x-operation on the &amp;lt;math&amp;gt; i\,\!&amp;lt;/math&amp;gt;th qubit and similarly for the other operators.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' Product of two Hilbert-Schmidt operators.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt operators:''' Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:''' Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:''' Property of certain binary operations that are possessed by Lie groups that can be written &amp;lt;math&amp;gt;&lt;br /&gt;
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:''' The symbol &amp;lt;math&amp;gt;\delta_{ij}\!&amp;lt;/math&amp;gt; that is defined as:&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\delta_{ij} = \begin{cases}&lt;br /&gt;
               1, &amp;amp; \mbox{if } i=j, \\&lt;br /&gt;
               0, &amp;amp; \mbox{if } i\neq j.&lt;br /&gt;
              \end{cases}&lt;br /&gt;
&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:''' A vector space, over some field with a binary operation, called a Lie bracket, that possesses bilinearity, is alternating on the vector space, and satisfies the Jacobi identity.&lt;br /&gt;
&lt;br /&gt;
'''Lie group:''' A differentiable manifold that corresponds to a continuous set of symmetries.&lt;br /&gt;
&lt;br /&gt;
'''Linear code:''' Error correcting code for which any linear combination of codewords is a codeword. Can be generated by a Generator matrix.&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:''' A very general mapping of a matrix acting on a vector that produces another vector. &lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:''' See ''Local operations''.&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Local operations:'''  Actions on an individual particle without involving any other particle.&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:''' Short for ''Logical qubit''. Qubits that are encoded with information that is to be protected from errors. Represented by &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; in &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; codes.&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2367</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2367"/>
		<updated>2013-10-17T19:42:26Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction Hamiltonian:''' When this Hamiltonian is between two qubits labelled &amp;lt;math&amp;gt; i \,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; j \,\!&amp;lt;/math&amp;gt;, it can be expressed as &amp;lt;math&amp;gt; &lt;br /&gt;
H_{ex}^{i,j} =  X_iX_j + Y_iY_j + Z_iZ_j,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt; X_i \,\!&amp;lt;/math&amp;gt; is the Pauli x-operation on the &amp;lt;math&amp;gt; i\,\!&amp;lt;/math&amp;gt;th qubit and similarly for the other operators.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' Product of two Hilbert-Schmidt operators.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt operators:''' Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:''' Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:''' Property of certain binary operations that are possessed by Lie groups that can be written &amp;lt;math&amp;gt;&lt;br /&gt;
f_{ilm}f_{jkl} + f_{jlm}f_{kil} + f_{klm}f_{ijl} =0,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2366</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2366"/>
		<updated>2013-10-17T19:18:14Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction Hamiltonian:''' When this Hamiltonian is between two qubits labelled &amp;lt;math&amp;gt; i \,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; j \,\!&amp;lt;/math&amp;gt;, it can be expressed as &amp;lt;math&amp;gt; &lt;br /&gt;
H_{ex}^{i,j} =  X_iX_j + Y_iY_j + Z_iZ_j,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt; X_i \,\!&amp;lt;/math&amp;gt; is the Pauli x-operation on the &amp;lt;math&amp;gt; i\,\!&amp;lt;/math&amp;gt;th qubit and similarly for the other operators.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' Product of two Hilbert-Schmidt operators.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt operators:''' Bounded, compact operators that are isometrically isomorphic to the tensor product of Hilbert spaces.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:''' Mathematical concept that makes vector operations and calculus possible in spaces that are greater than three dimensions.&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2365</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2365"/>
		<updated>2013-10-17T19:06:44Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:''' Restricts the rate of the error correcting code. A code that reaches the Hamming bound has perfect efficiency.&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:''' A &amp;lt;math&amp;gt;[7,4,3]\!&amp;lt;/math&amp;gt; code. This code, as the notation indicates, encodes &amp;lt;math&amp;gt;k = 4\!&amp;lt;/math&amp;gt; bits of information into &amp;lt;math&amp;gt;n = 7\!&amp;lt;/math&amp;gt; bits. One error can be detected and corrected at a distance of up to 3.&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:''' The number of places where two vectors differ.&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:''' The number of non-zero components of a vector or string.&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction:'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2364</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2364"/>
		<updated>2013-10-17T19:00:32Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:''' Set of objects that is closed under multiplication, associative, invertible and contain an identity element.&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:''' Quantum algorithm for searching unsorted databases. [http://www.quantiki.org/wiki/Grover's_search_algorithm See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' Also known as a Hadamard transformation and refers to the rotation of a single qubit used as the first step in many quantum algorithms.&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction:'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2363</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2363"/>
		<updated>2013-10-17T18:51:05Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Finite ''(or Galois)'' Field:''' A set of finite elements that is closed under vector addition and multiplication.&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:''' The set of invertible &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; matrices with complex numbers as entries.  It is denoted &amp;lt;math&amp;gt;GL(N,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''  Consider a discrete group (or subgroup of a larger group). There exists a subset of the group elements that will give all of the (sub)group elements through multiplication. The elements in this subset are the generators of the group.&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:''' An &amp;lt;math&amp;gt; n\times k\,\!&amp;lt;/math&amp;gt; matrix with columns that form a basis for the &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt;-dimensional coding sub-space of the &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-dimensional binary vector space. The vectors comprising the rows form a basis that will span the code space.&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:'''&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:''' &lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' An operator that corresponds with the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction:'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A homomorphism that is both injective and surjective.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2362</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2362"/>
		<updated>2013-10-17T18:35:41Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Field:''' &lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:'''&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' The Hamiltonian operator gives the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction (8.5.2):'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:''' The composition of two functions yields the product of those functions&amp;lt;math&amp;gt;&lt;br /&gt;
f(A\circ B) = f(A) \cdot f(B).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A one-to-one and onto mapping.  &lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2361</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2361"/>
		<updated>2013-10-17T17:23:04Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:''' Error which is symmetric in the three possible errors: bit flip, phase flip, and bit flip and phase flip.&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M\!&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}\!&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:''' A topological space that is close enough to Euclidean space to do calculus, often used in Group Theory. &lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:''' A function that is valued zero everywhere but at zero, where the function is a very thin spike of infinite height, and the area under that spike is 1.&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:''' A method of associating every code word with just one vector. In that case, when any errors occur, they can be traced back to their specific code word and corrected.&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:''' A quantum error correcting code of distance &amp;lt;math&amp;gt;d = 2t + 1\!&amp;lt;/math&amp;gt; can correct &amp;lt;math&amp;gt;t\!&amp;lt;/math&amp;gt; errors. Denoted by the value &amp;lt;math&amp;gt;d\!&amp;lt;/math&amp;gt; when a code is expressed in the form &amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:''' Set of criteria that are required for the physical system of a quantum computer. [http://www.quantiki.org/wiki/Quantum_computing See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual code:''' Denoted &amp;lt;math&amp;gt;\mathcal{C}^\perp\,\!&amp;lt;/math&amp;gt;, is the set of all vectors that have zero inner product with all &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  In other words, it is the set of all vectors &amp;lt;math&amp;gt;u\,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;u\cdot v = 0\,\!&amp;lt;/math&amp;gt; for all  &amp;lt;math&amp;gt;v\in \mathcal{C}\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:''' See ''Parity Check Matrix''.&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:''' Functions that are both orthogonal and linearly independent in quantum physics used to diagonalize matrices. They contain some multiplier, known as an eigenvalue.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' A nonzero solution to the eigenfunction that fulfills the initial conditions of the problem. This concept is most easily seen in the Schrodinger in its general form, &amp;lt;math&amp;gt;H|\psi \rangle = E|\psi \rangle\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue that transforms the Hamiltonian, &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H\!&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E\!&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y\!&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:''' A state that is not separable, where two particles interact on a quantum level and are then described as relative to one another. These states do not seem dependent on distance.&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:''' Named for its creators, Einstein, Podolsky, and Rosen. Paper that gave an example proving our incomplete knowledge of quantum mechanics, made famous in 1935.&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:''' Cyclic permutation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated})&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:''' Two representations &amp;lt;math&amp;gt;D^{(1)}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;D^{(2)} \,\!&amp;lt;/math&amp;gt; are equivalent if and only if there is an invertible matrix &amp;lt;math&amp;gt;S \,\!&amp;lt;/math&amp;gt; such that &amp;lt;math&amp;gt;D^{(1)} = SD^{(2)}S^{-1} \,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:''' Set of eigenvalues that give information about a given error, but not the code word that is affected by the error.&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:''' A three-dimensional rotational transformation that can be expressed as &amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt; Fully explained[[Appendix C - Vectors and Linear Algebra#Transformations of a Qubit| here.]]&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:''' The most probable outcome of a measurement in quantum physics.&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:''' A matrix representation of an abstract group that is also isomorphic to the set of elements in the abstract group&lt;br /&gt;
&lt;br /&gt;
'''Field:''' &lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:'''&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' The Hamiltonian operator gives the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction (8.5.2):'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A one-to-one and onto mapping.  &lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[[n,k,d]]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2360</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2360"/>
		<updated>2013-10-05T07:12:37Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[n,k,d]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:''' A group &amp;lt;math&amp;gt;G\!&amp;lt;/math&amp;gt; with elements that can be expressed as &amp;lt;math&amp;gt;g^n\!&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;n\in\mathbb{Z}\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;g\in G&amp;lt;/math&amp;gt;. Thus, &amp;lt;math&amp;gt;G= \{g^0,g^1,g^2,g^3,. . .\}\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Dagger (&amp;lt;math&amp;gt;\dagger\!&amp;lt;/math&amp;gt;):''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal in the Hamiltonian operator. Degenerate quantum codes have been studied because they contain many interesting characteristics.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:''' See ''Density operator''.&lt;br /&gt;
&lt;br /&gt;
'''Density operator:''' Language useful in describing a quantum system. In a pure state, the density operator &amp;lt;math&amp;gt; \rho\!&amp;lt;/math&amp;gt; can describe a quantum system whose state is known. For a state &amp;lt;math&amp;gt;|\psi\rangle\!&amp;lt;/math&amp;gt;, it can be said &amp;lt;math&amp;gt;\rho = |\psi\rangle\langle\psi|&amp;lt;/math&amp;gt;. In a mixed state, it can be described as a compilation of several pure states for &amp;lt;math&amp;gt;\rho\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:'''&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:'''&lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:'''&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:'''&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:'''  See Section &lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Dual of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:'''&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' &lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:'''&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:'''&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:'''&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:'''&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:'''&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:'''&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Field:'''&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:'''&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' The Hamiltonian operator gives the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction (8.5.2):'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A one-to-one and onto mapping.  &lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[n,k,d]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2359</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2359"/>
		<updated>2013-09-26T19:40:11Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[n,k,d]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:'''&lt;br /&gt;
&lt;br /&gt;
'''Dagger:''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:'''&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:'''&lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:'''&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:'''&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:'''  See Section &lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Dual of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:'''&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' &lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:'''&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:'''&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:'''&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:'''&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:'''&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:'''&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Field:'''&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:'''&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' The Hamiltonian operator gives the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction (8.5.2):'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A one-to-one and onto mapping.  &lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[n,k,d]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
'''Positive definite matrix:''' Matrix whose eigenvalues are all greater than zero. &lt;br /&gt;
&lt;br /&gt;
'''Semi-definite matrix:'''  Matrix whose eigenvalues are nonnegative.  &lt;br /&gt;
&lt;br /&gt;
'''Probability for existing in a state:'''&lt;br /&gt;
&lt;br /&gt;
'''Projector:''' A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Projection postulate:'''&lt;br /&gt;
&lt;br /&gt;
'''Pure state:'''&lt;br /&gt;
&lt;br /&gt;
'''QKD:''' See ''Quantum key distribution''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum bit:''' See ''Qubit''.&lt;br /&gt;
&lt;br /&gt;
'''Quantum cryptography:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum dense coding:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum gate:''' A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
'''Quantum hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum key distribution:'''&lt;br /&gt;
&lt;br /&gt;
'''Quantum NOT gate:''' One-bit gate, with one input and one output. If the input is 1, the output is 0, and vice versa.&lt;br /&gt;
&lt;br /&gt;
'''Qubit:''' A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Rank:'''&lt;br /&gt;
&lt;br /&gt;
'''Rate of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Reduced density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Representation space:'''&lt;br /&gt;
&lt;br /&gt;
'''Reversible quantum operation:'''  Operation for which every state on which the operator can act there exists an operation which restores it to its original state.&lt;br /&gt;
&lt;br /&gt;
'''RSA encryption:'''&lt;br /&gt;
&lt;br /&gt;
'''Schmidt decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''Schrodinger's Equation:'''&lt;br /&gt;
&lt;br /&gt;
'''Set:''' Any mathematical construct, often represented by a capital letter.&lt;br /&gt;
&lt;br /&gt;
'''Shor's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''Shor's nine-bit quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''Similarity transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular values:'''&lt;br /&gt;
&lt;br /&gt;
'''Singular value decomposition:'''&lt;br /&gt;
&lt;br /&gt;
'''SMR representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Special unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Spin:'''&lt;br /&gt;
&lt;br /&gt;
'''Spooky action at a distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizers of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Stabilizer code:'''&lt;br /&gt;
&lt;br /&gt;
'''Standard deviation:'''&lt;br /&gt;
&lt;br /&gt;
'''Stationary subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Stirling's formula:'''&lt;br /&gt;
&lt;br /&gt;
'''Subgroup:'''&lt;br /&gt;
&lt;br /&gt;
'''Superposition:''' A qubit state in superposition, where &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
'''Syndrome measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Taylor expansion:'''&lt;br /&gt;
&lt;br /&gt;
'''Teleportation:'''&lt;br /&gt;
&lt;br /&gt;
'''Tensor product:'''&lt;br /&gt;
&lt;br /&gt;
'''Trace:''' The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
'''Transpose:'''&lt;br /&gt;
&lt;br /&gt;
'''Trivial representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Turing machine:'''&lt;br /&gt;
&lt;br /&gt;
'''Uncertainty principle:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Unitary transformation:''' A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
'''Universal quantum computing:'''&lt;br /&gt;
&lt;br /&gt;
'''Universal set of gates:''' Universality. (2.6)&lt;br /&gt;
&lt;br /&gt;
'''Variance:'''&lt;br /&gt;
&lt;br /&gt;
'''Vector:''' A directed quantity.&lt;br /&gt;
&lt;br /&gt;
'''Vector space:'''&lt;br /&gt;
&lt;br /&gt;
'''Weight of a vector:''' See ''Hamming weight''.&lt;br /&gt;
&lt;br /&gt;
'''Weight of an operator:''' The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
'''Wigner-Clebsch-Gordon Coefficients:'''&lt;br /&gt;
&lt;br /&gt;
'''X gate:''' (2.3.2)&lt;br /&gt;
&lt;br /&gt;
'''Y gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Z gate:''' Phase-flip gate. (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2358</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2358"/>
		<updated>2013-09-26T05:59:11Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;[n,k,d]\!&amp;lt;/math&amp;gt; code:''' A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
'''Abelian group:''' A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
'''Abelian subgroup:''' See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
'''Adjoint (of a matrix):''' The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
'''Ancilla:'''  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
'''Angular momentum:''' A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
'''Anti-commutation:''' Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Basis:''' Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
'''Bath system:''' Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
'''Bell's theorem:''' &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
'''Bit flip error:'''  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
'''Bloch sphere:''' Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
'''Block diagonal matrix:'''  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
'''Bra-ket notation:''' Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Centralizer:'''  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Checksum:''' See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
'''Classical bit:''' A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
'''Closed quantum system:''' A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
'''Code:''' Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
'''Codewords:''' Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
'''Commutator:''' The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
'''Complex conjugate:''' Two expressions, consisting of a real number (&amp;lt;math&amp;gt;\alpha\!&amp;lt;/math&amp;gt; ) and imaginary number (&amp;lt;math&amp;gt;\beta i\!&amp;lt;/math&amp;gt;), where the &amp;lt;math&amp;gt;\beta\!&amp;lt;/math&amp;gt; component is of the same magnitude, but different sign. For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Complex number:''' A complex number has a real and imaginary quantity.  A complex number can be represented in the form &amp;lt;math&amp;gt;\alpha+\beta i\!&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;\alpha,\beta,c,\theta\in\mathbb{R}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Controlled not ''(CNOT gate)'':''' Quantum gate essential in quantum computing. It consists of two qubits, control and target, where the target bit is flipped if and only if the control gate is &amp;lt;math&amp;gt;\left\vert{1}\right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
'''Controlled operation:''' An operation on a state or set of states that is conditioned on another state or set of states. Can often be expressed with an &amp;quot;If-then&amp;quot; or &amp;quot;If-then-else&amp;quot; statement.&lt;br /&gt;
&lt;br /&gt;
'''Coset of a group:''' Used in group theory where &amp;lt;math&amp;gt;\mathcal{S}\,\!&amp;lt;/math&amp;gt; is a subgroup and &amp;lt;math&amp;gt;G\,\!&amp;lt;/math&amp;gt; be an element of the group &amp;lt;math&amp;gt;\mathcal{G}\,\!&amp;lt;/math&amp;gt;.  The left '''coset''' is a subset of the group &amp;lt;math&amp;gt;G\mathcal{S} = \{GS|S\in\mathcal{S} \}.\,\!&amp;lt;/math&amp;gt; One can similarly define the right coset. &lt;br /&gt;
&lt;br /&gt;
'''CSS codes:''' Short for ''Calderbank-Shor-Steane codes''. Class of quantum error-correcting codes that is a subclass of the group of stabilizer codes. Made up of two classical linear codes, &amp;lt;math&amp;gt; \mathcal{C}_1\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; \mathcal{C}_2\,\!&amp;lt;/math&amp;gt;, that are of the form &amp;lt;math&amp;gt;[ [n,k,d] ]\!&amp;lt;/math&amp;gt;. Protects against both phase- and bit-flips.&lt;br /&gt;
&lt;br /&gt;
'''Cyclic group:'''&lt;br /&gt;
&lt;br /&gt;
'''Dagger:''' See ''Hermitian Conjugate'', ''Adjoint (of a matrix)''&lt;br /&gt;
&lt;br /&gt;
'''Definite matrix:''' See ''Matrix Properties''.&lt;br /&gt;
&lt;br /&gt;
'''Degenerate:''' Having two or more eigenvalues that are equal.&lt;br /&gt;
&lt;br /&gt;
'''Density matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Density operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Depolarizing error:'''&lt;br /&gt;
&lt;br /&gt;
'''Determinant:''' When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
'''Diagonalizable:''' A matrix &amp;lt;math&amp;gt;M&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
'''Differentiable manifold:'''&lt;br /&gt;
&lt;br /&gt;
'''Dirac delta function:'''&lt;br /&gt;
&lt;br /&gt;
'''Dirac notation:''' See ''Bra-Ket Notation.''&lt;br /&gt;
&lt;br /&gt;
'''Disjointness condition:'''&lt;br /&gt;
&lt;br /&gt;
'''Distance of a quantum error correcting code:'''&lt;br /&gt;
&lt;br /&gt;
'''DiVincenzo's requirements for quantum computing:'''  See Section &lt;br /&gt;
&lt;br /&gt;
'''Dot product:''' The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed. &lt;br /&gt;
&lt;br /&gt;
'''Dual matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Dual of a code:'''&lt;br /&gt;
&lt;br /&gt;
'''Eigenfunction:'''&lt;br /&gt;
&lt;br /&gt;
'''Eigenvalue:''' &lt;br /&gt;
&lt;br /&gt;
'''Eigenvector:''' If &amp;lt;math&amp;gt;HY=EY&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
'''Entangled state:'''&lt;br /&gt;
&lt;br /&gt;
'''Environment:''' See ''Bath System''.&lt;br /&gt;
&lt;br /&gt;
'''EPR paradox:'''&lt;br /&gt;
&lt;br /&gt;
'''Epsilon tensor:'''&lt;br /&gt;
&lt;br /&gt;
'''Equivalent representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Error syndrome:'''&lt;br /&gt;
&lt;br /&gt;
'''Euler angle parametrization:'''&lt;br /&gt;
&lt;br /&gt;
'''Euler's Law:''' &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
'''Expectation value:'''&lt;br /&gt;
&lt;br /&gt;
'''Exponentiating a matrix:''' See ''Matrix Exponential''.&lt;br /&gt;
&lt;br /&gt;
'''Faithful representation:'''&lt;br /&gt;
&lt;br /&gt;
'''Field:'''&lt;br /&gt;
&lt;br /&gt;
'''Gate:''' See ''Quantum Gate''.&lt;br /&gt;
&lt;br /&gt;
'''General linear group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generators of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Generator matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Gram-Schmidt Decomposition:''' See ''Schmidt Decomposition''.&lt;br /&gt;
&lt;br /&gt;
'''Group:'''&lt;br /&gt;
&lt;br /&gt;
'''Grover's algorithm:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;h\!&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;):''' Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
'''Hadamard gate:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming bound:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming code:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamming weight:'''&lt;br /&gt;
&lt;br /&gt;
'''Hamiltonian:''' The Hamiltonian operator gives the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg exchange interaction (8.5.2):'''&lt;br /&gt;
&lt;br /&gt;
'''Heisenberg uncertainty principle:''' See ''uncertainty principle''.&lt;br /&gt;
&lt;br /&gt;
'''Hermitian:''' An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
'''Hermitian conjugate:''' The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
'''Hidden variable theory:''' See ''Local hidden variable theory''.&lt;br /&gt;
&lt;br /&gt;
'''Hilbert-Schmidt inner product:''' (2.4)&lt;br /&gt;
&lt;br /&gt;
'''Hilbert space:'''&lt;br /&gt;
&lt;br /&gt;
'''Homomorphism:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;i\!&amp;lt;/math&amp;gt;:'''  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
'''Identity matrix:''' A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
'''Inner product:''' See ''Dot product''.&lt;br /&gt;
&lt;br /&gt;
'''Inverse of a matrix:''' The inverse of a square matrix &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}\!&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = mathbb{I} = A^{-1}A \!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;mathbb{I}\!&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, only one or the other.)&lt;br /&gt;
&lt;br /&gt;
'''Invertible matrix:''' A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
'''Isolated system:''' See ''Closed system.''  &lt;br /&gt;
&lt;br /&gt;
'''Isomorphism:'''  A one-to-one and onto mapping.  &lt;br /&gt;
&lt;br /&gt;
'''Isotropy group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Isotropy subgroup:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Jacobi identity:'''&lt;br /&gt;
&lt;br /&gt;
'''Ket:''' See ''Bra-Ket Notation''.&lt;br /&gt;
&lt;br /&gt;
'''Kraus representation:''' Kraus decomposition. See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Kronecker delta:'''&lt;br /&gt;
&lt;br /&gt;
'''Levi-Civita symbol:''' See ''Epsilon tensor''.&lt;br /&gt;
&lt;br /&gt;
'''Lie algebra:'''&lt;br /&gt;
&lt;br /&gt;
'''Lie group:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear code:'''&lt;br /&gt;
&lt;br /&gt;
'''Linear combination:''' A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
'''Linear map:'''&lt;br /&gt;
&lt;br /&gt;
'''Little group:''' See ''Stabilizer''.&lt;br /&gt;
&lt;br /&gt;
'''Local actions:'''&lt;br /&gt;
&lt;br /&gt;
'''Local hidden variable theory:''' See ''hidden variable theory''&lt;br /&gt;
&lt;br /&gt;
'''Logical bit:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix exponential:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix properties:'''&lt;br /&gt;
&lt;br /&gt;
'''Matrix transformation:'''&lt;br /&gt;
&lt;br /&gt;
'''Measurement:'''&lt;br /&gt;
&lt;br /&gt;
'''Minimum distance:'''&lt;br /&gt;
&lt;br /&gt;
'''Modular arithmetic:''' When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
'''Modulus:'''&lt;br /&gt;
&lt;br /&gt;
'''&amp;lt;math&amp;gt;n,k,d\!&amp;lt;/math&amp;gt; code:''' See ''&amp;lt;math&amp;gt;[n,k,d]\!&amp;lt;/math&amp;gt; code.''&lt;br /&gt;
&lt;br /&gt;
'''No cloning theorem:''' There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
'''Noise:'''&lt;br /&gt;
&lt;br /&gt;
'''Non-degenerate code:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalizer:'''&lt;br /&gt;
&lt;br /&gt;
'''Normalization:''' A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
'''One-to-one:''' A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
'''Onto:''' A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
'''Open system:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator:'''&lt;br /&gt;
&lt;br /&gt;
'''Operator-sum representation:''' See ''SMR representation''.&lt;br /&gt;
&lt;br /&gt;
'''Order of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Ordered basis:'''&lt;br /&gt;
&lt;br /&gt;
'''Orthogonal:''' Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
'''Outer product:'''&lt;br /&gt;
&lt;br /&gt;
'''P gate ''(Not phase gate)'':'''&lt;br /&gt;
&lt;br /&gt;
'''Parity:'''&lt;br /&gt;
&lt;br /&gt;
'''Parity check:''' See ''Inner product''.&lt;br /&gt;
&lt;br /&gt;
'''Parity check matrix:'''&lt;br /&gt;
&lt;br /&gt;
'''Partial trace:'''&lt;br /&gt;
&lt;br /&gt;
'''Partition of a group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli group:'''&lt;br /&gt;
&lt;br /&gt;
'''Pauli matrices:''' The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
'''Permutation:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase flip error:'''&lt;br /&gt;
&lt;br /&gt;
'''Phase gate:''' See Z gate&lt;br /&gt;
&lt;br /&gt;
'''Planck's constant:'''&lt;br /&gt;
&lt;br /&gt;
'''Polarization:'''&lt;br /&gt;
&lt;br /&gt;
Positive definite semidefinite matrix:  A positive definite matrix is one whose eigenvalues are all greater than zero.  A positive semidefinite matrix has no negative eigenvalues.  &lt;br /&gt;
&lt;br /&gt;
Probability for existing in a state:&lt;br /&gt;
&lt;br /&gt;
Projector: A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
Projection postulate&lt;br /&gt;
&lt;br /&gt;
Pure state&lt;br /&gt;
&lt;br /&gt;
QKD: See quantum key distribution&lt;br /&gt;
&lt;br /&gt;
Quantum bit: See Qubit&lt;br /&gt;
&lt;br /&gt;
Quantum cryptography&lt;br /&gt;
&lt;br /&gt;
Quantum dense coding&lt;br /&gt;
&lt;br /&gt;
Quantum gate: A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
Quantum hamming bound&lt;br /&gt;
&lt;br /&gt;
Quantum key distribution:&lt;br /&gt;
&lt;br /&gt;
Quantum NOT gate: see X gate, but be careful, the NOT gate is only defined for qubits.&lt;br /&gt;
&lt;br /&gt;
Qubit: A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
Rank&lt;br /&gt;
&lt;br /&gt;
Rate of a code&lt;br /&gt;
&lt;br /&gt;
Reduced density operator&lt;br /&gt;
&lt;br /&gt;
Representation space&lt;br /&gt;
&lt;br /&gt;
Reversible  quantum operation:  An operation is reversible if for every state on which the operator can act, there exists an operation which restores the state to its original.&lt;br /&gt;
&lt;br /&gt;
RSA encryption&lt;br /&gt;
&lt;br /&gt;
Schmidt decomposition&lt;br /&gt;
&lt;br /&gt;
Schrodinger's Equation&lt;br /&gt;
&lt;br /&gt;
Set: Any mathematical construct.&lt;br /&gt;
&lt;br /&gt;
Shor's algorithm&lt;br /&gt;
&lt;br /&gt;
Shor's nine-bit quantum error correcting code&lt;br /&gt;
&lt;br /&gt;
Similarity transformation&lt;br /&gt;
&lt;br /&gt;
Singular values&lt;br /&gt;
&lt;br /&gt;
Singular value decomposition&lt;br /&gt;
&lt;br /&gt;
SMR representation&lt;br /&gt;
&lt;br /&gt;
Special unitary matrix&lt;br /&gt;
&lt;br /&gt;
Spin&lt;br /&gt;
&lt;br /&gt;
Spooky action at a distance&lt;br /&gt;
&lt;br /&gt;
Stabilizers of a group&lt;br /&gt;
&lt;br /&gt;
Stabilizer code&lt;br /&gt;
&lt;br /&gt;
Standard deviation&lt;br /&gt;
&lt;br /&gt;
Stationary subgroup (see stabilizer)&lt;br /&gt;
&lt;br /&gt;
Stirling's formula&lt;br /&gt;
&lt;br /&gt;
Subgroup&lt;br /&gt;
&lt;br /&gt;
Superposition: A qubit state in superposition, &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
Syndrome measurement&lt;br /&gt;
&lt;br /&gt;
Taylor expansion&lt;br /&gt;
&lt;br /&gt;
Teleportation&lt;br /&gt;
&lt;br /&gt;
Tensor product&lt;br /&gt;
&lt;br /&gt;
Trace: The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
Transpose&lt;br /&gt;
&lt;br /&gt;
Trivial representation&lt;br /&gt;
&lt;br /&gt;
Turing machine&lt;br /&gt;
&lt;br /&gt;
Uncertainty principle&lt;br /&gt;
&lt;br /&gt;
Unitary matrix&lt;br /&gt;
&lt;br /&gt;
Unitary transformation: A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
Universal quantum computing&lt;br /&gt;
&lt;br /&gt;
Universal set of gates (universality) (2.6)&lt;br /&gt;
&lt;br /&gt;
Variance&lt;br /&gt;
&lt;br /&gt;
Vector: A directed quantity.&lt;br /&gt;
&lt;br /&gt;
Vector space&lt;br /&gt;
&lt;br /&gt;
Weight of a vector (see Hamming weight)&lt;br /&gt;
&lt;br /&gt;
Weight of an operator: The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
Wigner-Clebsch-Gordon Coefficients&lt;br /&gt;
&lt;br /&gt;
X gate (2.3.2)&lt;br /&gt;
&lt;br /&gt;
Y gate&lt;br /&gt;
&lt;br /&gt;
Z gate, or phase-flip gate (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2357</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2357"/>
		<updated>2013-09-23T03:09:52Z</updated>

		<summary type="html">&lt;p&gt;Hilary: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;[ [n,k,d] ]&amp;lt;/math&amp;gt; code: A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
Abelian group: A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
Abelian subgroup: See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
Adjoint (of a matrix): The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
Ancilla:  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
Angular momentum: A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
Anti-commutation:  Two operators anti-commute when &amp;lt;math&amp;gt;AB+BA=0\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Basis: Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
Bath system: Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
Bell's theorem: &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
Bit flip error:  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
Bloch sphere: Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
Block diagonal matrix:  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
Bra-ket notation: Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Centralizer:  Subgroup of a group. Consists of elements of the group that commute with all elements of a certain set. For some &amp;lt;math&amp;gt;v \in S&amp;lt;/math&amp;gt; and all &amp;lt;math&amp;gt;c \in S&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;vc = cv\!&amp;lt;/math&amp;gt;, for some group &amp;lt;math&amp;gt;S\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
Checksum: See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
Classical bit: A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
Closed quantum system: A quantum system that experiences no unwanted interference. The universe as a whole is viewed as a closed system.&lt;br /&gt;
&lt;br /&gt;
Code: Short for ''Quantum Code''. Used in correcting errors in quantum systems. Where &amp;lt;math&amp;gt;[n,k]\!&amp;lt;/math&amp;gt; is the way we describe a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt; with &amp;lt;math&amp;gt;n\!&amp;lt;/math&amp;gt; bits that are used to encode &amp;lt;math&amp;gt;k\!&amp;lt;/math&amp;gt; bits.&lt;br /&gt;
&lt;br /&gt;
Codewords: Used to describe the set of all elements in a code &amp;lt;math&amp;gt;C\!&amp;lt;/math&amp;gt;. There are &amp;lt;math&amp;gt;2^n\,\!&amp;lt;/math&amp;gt; &amp;lt;math&amp;gt;n\,\!&amp;lt;/math&amp;gt;-bit words in the space.&lt;br /&gt;
&lt;br /&gt;
Commutator: The commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]\!&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA\!&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\!&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
Complex conjugate: For some &amp;lt;math&amp;gt;\alpha, \beta \in \mathbb{R}, \alpha i + \beta&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\alpha i - \beta\!&amp;lt;/math&amp;gt; are complex conjugates.&lt;br /&gt;
&lt;br /&gt;
Complex number: A complex number has a real and imaginary part.  A complex number can be represented in the form &amp;lt;math&amp;gt;a+bi&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
Controlled not (CNOT gate):  &lt;br /&gt;
&lt;br /&gt;
Controlled operation: An operation on a state or set of states that is conditioned on another state or set of states.&lt;br /&gt;
&lt;br /&gt;
Coset of a group&lt;br /&gt;
&lt;br /&gt;
CSS codes&lt;br /&gt;
&lt;br /&gt;
Cyclic group&lt;br /&gt;
&lt;br /&gt;
Dagger (see hermitian conjugate or adjoint)&lt;br /&gt;
&lt;br /&gt;
Definite matrix (see matrix properties)&lt;br /&gt;
&lt;br /&gt;
Degenerate: Having two or more eigenvalues that are equal.&lt;br /&gt;
&lt;br /&gt;
Density matrix&lt;br /&gt;
&lt;br /&gt;
Density operator&lt;br /&gt;
&lt;br /&gt;
Depolarizing error&lt;br /&gt;
&lt;br /&gt;
Determinant: When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
Diagonalizable: A matrix &amp;lt;math&amp;gt;M&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
Differentiable manifold&lt;br /&gt;
&lt;br /&gt;
Dirac delta function&lt;br /&gt;
&lt;br /&gt;
Dirac notation (see bra-ket notation)&lt;br /&gt;
&lt;br /&gt;
Disjointness condition&lt;br /&gt;
&lt;br /&gt;
Distance of a quantum error correcting code&lt;br /&gt;
&lt;br /&gt;
DiVincenzo's requirements for quantum computing:  See Section &lt;br /&gt;
&lt;br /&gt;
Dot product: The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed.  &lt;br /&gt;
&lt;br /&gt;
Dual matrix&lt;br /&gt;
&lt;br /&gt;
Dual of a code&lt;br /&gt;
&lt;br /&gt;
Eigenfunction, eigenvalue, eigenvector: If &amp;lt;math&amp;gt;HY=EY&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
Entangled state&lt;br /&gt;
&lt;br /&gt;
Environment (see Bath system)&lt;br /&gt;
&lt;br /&gt;
EPR paradox&lt;br /&gt;
&lt;br /&gt;
Epsilon tensor&lt;br /&gt;
&lt;br /&gt;
Equivalent representation&lt;br /&gt;
&lt;br /&gt;
Error syndrome&lt;br /&gt;
&lt;br /&gt;
Euler angle parametrization&lt;br /&gt;
&lt;br /&gt;
Euler's law: &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Expectation value:&lt;br /&gt;
&lt;br /&gt;
Exponentiating a matrix (see matrix exponential)&lt;br /&gt;
&lt;br /&gt;
Faithful representation&lt;br /&gt;
&lt;br /&gt;
Field&lt;br /&gt;
&lt;br /&gt;
Gate (see Quantum gate)&lt;br /&gt;
&lt;br /&gt;
General linear group&lt;br /&gt;
&lt;br /&gt;
Generators of a group&lt;br /&gt;
&lt;br /&gt;
Generator matrix&lt;br /&gt;
&lt;br /&gt;
Gram-Schmidt decomposition (see Schmidt decomposition)&lt;br /&gt;
&lt;br /&gt;
Group&lt;br /&gt;
&lt;br /&gt;
Grover's algorithm&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;): Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
Hadamard gate&lt;br /&gt;
&lt;br /&gt;
Hamming bound&lt;br /&gt;
&lt;br /&gt;
Hamming code&lt;br /&gt;
&lt;br /&gt;
Hamming distance&lt;br /&gt;
&lt;br /&gt;
Hamming weight&lt;br /&gt;
&lt;br /&gt;
Hamiltonian: The Hamiltonian operator gives the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
Heisenberg exchange interaction (8.5.2):&lt;br /&gt;
&lt;br /&gt;
Heisenberg uncertainty principle (see uncertainty principle)&lt;br /&gt;
&lt;br /&gt;
Hermitian: An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
Hermitian conjugate: The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
Hidden variable theory (see also local hidden variable theory):&lt;br /&gt;
&lt;br /&gt;
Hilbert-Schmidt inner product (2.4)&lt;br /&gt;
&lt;br /&gt;
Hilbert space&lt;br /&gt;
&lt;br /&gt;
Homomorphism&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;:  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
Identity matrix: A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
Inner product (see dot product)&lt;br /&gt;
&lt;br /&gt;
Inverse of a matrix: The inverse of a square matrix &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = I = A^{-1}A &amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, but only one or the other.)&lt;br /&gt;
&lt;br /&gt;
Invertible matrix: A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
Isolated system (see Closed system):  A system which does not interact with any other system.  &lt;br /&gt;
&lt;br /&gt;
Isomorphism:  A one-to-one and onto mapping.  &lt;br /&gt;
&lt;br /&gt;
Isotropy group or Isotropy subgroup (see stabilizer)&lt;br /&gt;
&lt;br /&gt;
Jacobi identity&lt;br /&gt;
&lt;br /&gt;
Ket: See bra-ket notation&lt;br /&gt;
&lt;br /&gt;
Kraus representation (or Kraus decomposition) (see SMR representation)&lt;br /&gt;
&lt;br /&gt;
Kronecker delta&lt;br /&gt;
&lt;br /&gt;
Levi-Civita symbol (see epsilon tensor)&lt;br /&gt;
&lt;br /&gt;
Lie algebra&lt;br /&gt;
&lt;br /&gt;
Lie group&lt;br /&gt;
&lt;br /&gt;
Linear code&lt;br /&gt;
&lt;br /&gt;
Linear combination: A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
Linear map: &lt;br /&gt;
&lt;br /&gt;
Little group (see stabilizer)&lt;br /&gt;
&lt;br /&gt;
Local actions&lt;br /&gt;
&lt;br /&gt;
Local hidden variable theory (see also hidden variable theory):&lt;br /&gt;
&lt;br /&gt;
Logical bit&lt;br /&gt;
&lt;br /&gt;
Matrix exponential&lt;br /&gt;
&lt;br /&gt;
Matrix properties&lt;br /&gt;
&lt;br /&gt;
Matrix transformation&lt;br /&gt;
&lt;br /&gt;
Measurement&lt;br /&gt;
&lt;br /&gt;
Minimum distance&lt;br /&gt;
&lt;br /&gt;
Modular arithmetic: When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
Modulus&lt;br /&gt;
&lt;br /&gt;
n,k,d code (see [n,k,d] code)&lt;br /&gt;
&lt;br /&gt;
No cloning theorem: There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
Noise&lt;br /&gt;
&lt;br /&gt;
Non-degenerate code&lt;br /&gt;
&lt;br /&gt;
Normalizer:&lt;br /&gt;
&lt;br /&gt;
Normalization: A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
One-to-one: A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
Onto: A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
Open system&lt;br /&gt;
&lt;br /&gt;
Operator&lt;br /&gt;
&lt;br /&gt;
Operator-sum representation (see SMR representation)&lt;br /&gt;
&lt;br /&gt;
Order of a group&lt;br /&gt;
&lt;br /&gt;
Ordered basis&lt;br /&gt;
&lt;br /&gt;
Orthogonal: Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
Outer product&lt;br /&gt;
&lt;br /&gt;
P gate (not the phase gate):&lt;br /&gt;
&lt;br /&gt;
Parity&lt;br /&gt;
&lt;br /&gt;
Parity check (see inner product)&lt;br /&gt;
&lt;br /&gt;
Parity check matrix&lt;br /&gt;
&lt;br /&gt;
Partial trace&lt;br /&gt;
&lt;br /&gt;
Partition of a group&lt;br /&gt;
&lt;br /&gt;
Pauli group&lt;br /&gt;
&lt;br /&gt;
Pauli matrices: The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
Permutation:  &lt;br /&gt;
&lt;br /&gt;
Phase flip error&lt;br /&gt;
&lt;br /&gt;
Phase gate: See Z gate&lt;br /&gt;
&lt;br /&gt;
Planck's constant:&lt;br /&gt;
&lt;br /&gt;
Polarization&lt;br /&gt;
&lt;br /&gt;
Positive definite and semidefinite matrix:  A positive definite matrix is one whose eigenvalues are all greater than zero.  A positive semidefinite matrix has no negative eigenvalues.  &lt;br /&gt;
&lt;br /&gt;
Probability for existing in a state:&lt;br /&gt;
&lt;br /&gt;
Projector: A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
Projection postulate&lt;br /&gt;
&lt;br /&gt;
Pure state&lt;br /&gt;
&lt;br /&gt;
QKD: See quantum key distribution&lt;br /&gt;
&lt;br /&gt;
Quantum bit: See Qubit&lt;br /&gt;
&lt;br /&gt;
Quantum cryptography&lt;br /&gt;
&lt;br /&gt;
Quantum dense coding&lt;br /&gt;
&lt;br /&gt;
Quantum gate: A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
Quantum hamming bound&lt;br /&gt;
&lt;br /&gt;
Quantum key distribution:&lt;br /&gt;
&lt;br /&gt;
Quantum NOT gate: see X gate, but be careful, the NOT gate is only defined for qubits.&lt;br /&gt;
&lt;br /&gt;
Qubit: A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
Rank&lt;br /&gt;
&lt;br /&gt;
Rate of a code&lt;br /&gt;
&lt;br /&gt;
Reduced density operator&lt;br /&gt;
&lt;br /&gt;
Representation space&lt;br /&gt;
&lt;br /&gt;
Reversible  quantum operation:  An operation is reversible if for every state on which the operator can act, there exists an operation which restores the state to its original.&lt;br /&gt;
&lt;br /&gt;
RSA encryption&lt;br /&gt;
&lt;br /&gt;
Schmidt decomposition&lt;br /&gt;
&lt;br /&gt;
Schrodinger's Equation&lt;br /&gt;
&lt;br /&gt;
Set: Any mathematical construct.&lt;br /&gt;
&lt;br /&gt;
Shor's algorithm&lt;br /&gt;
&lt;br /&gt;
Shor's nine-bit quantum error correcting code&lt;br /&gt;
&lt;br /&gt;
Similarity transformation&lt;br /&gt;
&lt;br /&gt;
Singular values&lt;br /&gt;
&lt;br /&gt;
Singular value decomposition&lt;br /&gt;
&lt;br /&gt;
SMR representation&lt;br /&gt;
&lt;br /&gt;
Special unitary matrix&lt;br /&gt;
&lt;br /&gt;
Spin&lt;br /&gt;
&lt;br /&gt;
Spooky action at a distance&lt;br /&gt;
&lt;br /&gt;
Stabilizers of a group&lt;br /&gt;
&lt;br /&gt;
Stabilizer code&lt;br /&gt;
&lt;br /&gt;
Standard deviation&lt;br /&gt;
&lt;br /&gt;
Stationary subgroup (see stabilizer)&lt;br /&gt;
&lt;br /&gt;
Stirling's formula&lt;br /&gt;
&lt;br /&gt;
Subgroup&lt;br /&gt;
&lt;br /&gt;
Superposition: A qubit state in superposition, &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
Syndrome measurement&lt;br /&gt;
&lt;br /&gt;
Taylor expansion&lt;br /&gt;
&lt;br /&gt;
Teleportation&lt;br /&gt;
&lt;br /&gt;
Tensor product&lt;br /&gt;
&lt;br /&gt;
Trace: The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
Transpose&lt;br /&gt;
&lt;br /&gt;
Trivial representation&lt;br /&gt;
&lt;br /&gt;
Turing machine&lt;br /&gt;
&lt;br /&gt;
Uncertainty principle&lt;br /&gt;
&lt;br /&gt;
Unitary matrix&lt;br /&gt;
&lt;br /&gt;
Unitary transformation: A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
Universal quantum computing&lt;br /&gt;
&lt;br /&gt;
Universal set of gates (universality) (2.6)&lt;br /&gt;
&lt;br /&gt;
Variance&lt;br /&gt;
&lt;br /&gt;
Vector: A directed quantity.&lt;br /&gt;
&lt;br /&gt;
Vector space&lt;br /&gt;
&lt;br /&gt;
Weight of a vector (see Hamming weight)&lt;br /&gt;
&lt;br /&gt;
Weight of an operator: The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
Wigner-Clebsch-Gordon Coefficients&lt;br /&gt;
&lt;br /&gt;
X gate (2.3.2)&lt;br /&gt;
&lt;br /&gt;
Y gate&lt;br /&gt;
&lt;br /&gt;
Z gate, or phase-flip gate (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2356</id>
		<title>Glossary</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Glossary&amp;diff=2356"/>
		<updated>2013-09-20T00:41:28Z</updated>

		<summary type="html">&lt;p&gt;Hilary: adding to glossary&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&amp;lt;!-- Let us define things here or at least add links to definitions. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;[ [n,k,d] ]&amp;lt;/math&amp;gt; code: A code that uses ''n'' physical qubits to encode ''k'' logical qubits and will correct &amp;lt;math&amp;gt;\frac{(d-1)}{2}&amp;lt;/math&amp;gt; errors.&lt;br /&gt;
&lt;br /&gt;
Abelian group: A group for which all elements commute.  &lt;br /&gt;
&lt;br /&gt;
Abelian subgroup: See ''Abelian Group'', ''Subgroup''&lt;br /&gt;
&lt;br /&gt;
Adjoint (of a matrix): The transpose complex conjugate of an operator.  (Also referred to as the conjugate or Hermitian conjugate.)&lt;br /&gt;
&lt;br /&gt;
Ancilla:  An ancillary (or extra) qubit not used directly in the computation, but used for other purposes, e.g. error correction.  &lt;br /&gt;
&lt;br /&gt;
Angular momentum: A vector quantity classically defined as the cross product of the instantaneous position vector and instantaneous linear momentum. In quantum mechanics, it is defined as the cross product of the position and momentum operators.&lt;br /&gt;
&lt;br /&gt;
Anti-commutation:  Two operators anti-commute when AB+BA=0.&lt;br /&gt;
&lt;br /&gt;
Basis: Subset of a given vector space, V, that is both linearly independent and spans the vector space V. The number of elements in any basis are equal to the dimension of V.&lt;br /&gt;
&lt;br /&gt;
Bath system: Describes a system that has had an unwanted interaction with an open quantum system. Environment.&lt;br /&gt;
&lt;br /&gt;
Bell's theorem: &amp;quot;No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.&amp;quot; [http://www.quantiki.org/wiki/Bell's_theorem See Full Article]&lt;br /&gt;
&lt;br /&gt;
Bit flip error:  An error which takes &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left| 1 \right\rangle &amp;lt;/math&amp;gt; to &amp;lt;math&amp;gt;\left| 0 \right\rangle &amp;lt;/math&amp;gt;.  The operator which does this is the Pauli operator &amp;lt;math&amp;gt;\sigma_x&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
Bloch sphere: Unit sphere where a qubit can be represented as a point; geometrical representation of a two-level quantum system. [http://www.quantiki.org/wiki/File:Bloch.png Image]&lt;br /&gt;
&lt;br /&gt;
Block diagonal matrix:  A matrix which has non-zero elements only in blocks along the diagonal.  &lt;br /&gt;
&lt;br /&gt;
Bra-ket notation: Used in quantum mechanics to describe quantum states, consisting of two distinct symbols: the left symbol, ''bra'', and the right symbol, ''ket''. ''Bra:'' &amp;lt;math&amp;gt;\left\langle\phi\right\vert&amp;lt;/math&amp;gt;; ''Ket:'' &amp;lt;math&amp;gt;\left\vert\psi\right\rangle&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Centralizer:  &lt;br /&gt;
&lt;br /&gt;
Checksum: See ''Dot Product''&lt;br /&gt;
&lt;br /&gt;
Classical bit: A classical bit is represented by two different states of a classical system, which are represented by 1 and 0. (1.3)&lt;br /&gt;
&lt;br /&gt;
Closed system:  &lt;br /&gt;
&lt;br /&gt;
Code:  &lt;br /&gt;
&lt;br /&gt;
Codewords:  &lt;br /&gt;
&lt;br /&gt;
Commutator: The commutator of &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B&amp;lt;/math&amp;gt; is denoted &amp;lt;math&amp;gt;[A,B]&amp;lt;/math&amp;gt;, which means &amp;lt;math&amp;gt;AB-BA&amp;lt;/math&amp;gt;.  Its value may be found by implementing the operators of &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B&amp;lt;/math&amp;gt; on a test function.  If the commutator of &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B&amp;lt;/math&amp;gt; is zero, they are said to commute.&lt;br /&gt;
&lt;br /&gt;
Complex conjugate:&lt;br /&gt;
&lt;br /&gt;
Complex number: A complex number has a real and imaginary part.  A complex number can be represented in the form &amp;lt;math&amp;gt;a+bi&amp;lt;/math&amp;gt; or, &amp;lt;math&amp;gt;ce^{(i\theta)}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
Controlled not (CNOT gate):  &lt;br /&gt;
&lt;br /&gt;
Controlled operation: An operation on a state or set of states that is conditioned on another state or set of states.&lt;br /&gt;
&lt;br /&gt;
Coset of a group&lt;br /&gt;
&lt;br /&gt;
CSS codes&lt;br /&gt;
&lt;br /&gt;
Cyclic group&lt;br /&gt;
&lt;br /&gt;
Dagger (see hermitian conjugate or adjoint)&lt;br /&gt;
&lt;br /&gt;
Definite matrix (see matrix properties)&lt;br /&gt;
&lt;br /&gt;
Degenerate: Having two or more eigenvalues that are equal.&lt;br /&gt;
&lt;br /&gt;
Density matrix&lt;br /&gt;
&lt;br /&gt;
Density operator&lt;br /&gt;
&lt;br /&gt;
Depolarizing error&lt;br /&gt;
&lt;br /&gt;
Determinant: When rows or columns of a matrix are taken as vectors, the determinant is the volume enclosed by those vectors and corresponding parallel vectors creating parallelograms.  Determinants only exist for square matrices.&lt;br /&gt;
&lt;br /&gt;
Diagonalizable: A matrix &amp;lt;math&amp;gt;M&amp;lt;/math&amp;gt; is diagonalizable when it can be put into the form &amp;lt;math&amp;gt;D=S^{-1}MS&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;S&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;S^{-1}&amp;lt;/math&amp;gt; exist and are inverses.&lt;br /&gt;
&lt;br /&gt;
Differentiable manifold&lt;br /&gt;
&lt;br /&gt;
Dirac delta function&lt;br /&gt;
&lt;br /&gt;
Dirac notation (see bra-ket notation)&lt;br /&gt;
&lt;br /&gt;
Disjointness condition&lt;br /&gt;
&lt;br /&gt;
Distance of a quantum error correcting code&lt;br /&gt;
&lt;br /&gt;
DiVincenzo's requirements for quantum computing:  See Section &lt;br /&gt;
&lt;br /&gt;
Dot product: The scalar that results when two vectors have their corresponding components multiplied, and each of these products summed.  &lt;br /&gt;
&lt;br /&gt;
Dual matrix&lt;br /&gt;
&lt;br /&gt;
Dual of a code&lt;br /&gt;
&lt;br /&gt;
Eigenfunction, eigenvalue, eigenvector: If &amp;lt;math&amp;gt;HY=EY&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;H&amp;lt;/math&amp;gt; is a matrix, &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is a scalar and &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a vector, then &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is the eigenvector and &amp;lt;math&amp;gt;E&amp;lt;/math&amp;gt; is the eigenvalue. If &amp;lt;math&amp;gt;Y&amp;lt;/math&amp;gt; is a function, it is called an eigenfunction.&lt;br /&gt;
&lt;br /&gt;
Entangled state&lt;br /&gt;
&lt;br /&gt;
Environment (see Bath system)&lt;br /&gt;
&lt;br /&gt;
EPR paradox&lt;br /&gt;
&lt;br /&gt;
Epsilon tensor&lt;br /&gt;
&lt;br /&gt;
Equivalent representation&lt;br /&gt;
&lt;br /&gt;
Error syndrome&lt;br /&gt;
&lt;br /&gt;
Euler angle parametrization&lt;br /&gt;
&lt;br /&gt;
Euler's law: &amp;lt;math&amp;gt;\sin(x)+i\cos(x)=e^{ix}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Expectation value:&lt;br /&gt;
&lt;br /&gt;
Exponentiating a matrix (see matrix exponential)&lt;br /&gt;
&lt;br /&gt;
Faithful representation&lt;br /&gt;
&lt;br /&gt;
Field&lt;br /&gt;
&lt;br /&gt;
Gate (see Quantum gate)&lt;br /&gt;
&lt;br /&gt;
General linear group&lt;br /&gt;
&lt;br /&gt;
Generators of a group&lt;br /&gt;
&lt;br /&gt;
Generator matrix&lt;br /&gt;
&lt;br /&gt;
Gram-Schmidt decomposition (see Schmidt decomposition)&lt;br /&gt;
&lt;br /&gt;
Group&lt;br /&gt;
&lt;br /&gt;
Grover's algorithm&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; bar (&amp;lt;math&amp;gt;\hbar&amp;lt;/math&amp;gt;): Planck's constant divided by &amp;lt;math&amp;gt;2\pi&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;h&amp;lt;/math&amp;gt; is Planck's constant&lt;br /&gt;
&lt;br /&gt;
Hadamard gate&lt;br /&gt;
&lt;br /&gt;
Hamming bound&lt;br /&gt;
&lt;br /&gt;
Hamming code&lt;br /&gt;
&lt;br /&gt;
Hamming distance&lt;br /&gt;
&lt;br /&gt;
Hamming weight&lt;br /&gt;
&lt;br /&gt;
Hamiltonian: The Hamiltonian operator gives the total energy of the system.&lt;br /&gt;
&lt;br /&gt;
Heisenberg exchange interaction (8.5.2):&lt;br /&gt;
&lt;br /&gt;
Heisenberg uncertainty principle (see uncertainty principle)&lt;br /&gt;
&lt;br /&gt;
Hermitian: An operator whose transpose equals its complex conjugate, i.e., the matrix is equal to its adjoint.  &lt;br /&gt;
&lt;br /&gt;
Hermitian conjugate: The transpose complex conjugate of an operator.&lt;br /&gt;
&lt;br /&gt;
Hidden variable theory (see also local hidden variable theory):&lt;br /&gt;
&lt;br /&gt;
Hilbert-Schmidt inner product (2.4)&lt;br /&gt;
&lt;br /&gt;
Hilbert space&lt;br /&gt;
&lt;br /&gt;
Homomorphism&lt;br /&gt;
&lt;br /&gt;
&amp;lt;math&amp;gt;i&amp;lt;/math&amp;gt;:  Denotes the square root of negative one&lt;br /&gt;
&lt;br /&gt;
Identity matrix: A matrix of zeros except for the diagonal, where each element is 1.  Multiplying any matrix of the same dimension by it leaves the original matrix unchanged.&lt;br /&gt;
&lt;br /&gt;
Inner product (see dot product)&lt;br /&gt;
&lt;br /&gt;
Inverse of a matrix: The inverse of a square matrix &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; is the matrix, denoted &amp;lt;math&amp;gt;A^{-1}&amp;lt;/math&amp;gt;, such that &amp;lt;math&amp;gt;AA^{-1} = I = A^{-1}A &amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;I&amp;lt;/math&amp;gt; is identity matrix.  (When the matrix is not square, it is possible to have a left and/or right inverse which does not satisfy both of these relations, but only one or the other.)&lt;br /&gt;
&lt;br /&gt;
Invertible matrix: A matrix for which an inverse exists.&lt;br /&gt;
&lt;br /&gt;
Isolated system (see Closed system):  A system which does not interact with any other system.  &lt;br /&gt;
&lt;br /&gt;
Isomorphism:  A one-to-one and onto mapping.  &lt;br /&gt;
&lt;br /&gt;
Isotropy group or Isotropy subgroup (see stabilizer)&lt;br /&gt;
&lt;br /&gt;
Jacobi identity&lt;br /&gt;
&lt;br /&gt;
Ket: See bra-ket notation&lt;br /&gt;
&lt;br /&gt;
Kraus representation (or Kraus decomposition) (see SMR representation)&lt;br /&gt;
&lt;br /&gt;
Kronecker delta&lt;br /&gt;
&lt;br /&gt;
Levi-Civita symbol (see epsilon tensor)&lt;br /&gt;
&lt;br /&gt;
Lie algebra&lt;br /&gt;
&lt;br /&gt;
Lie group&lt;br /&gt;
&lt;br /&gt;
Linear code&lt;br /&gt;
&lt;br /&gt;
Linear combination: A set of vectors each multiplied by a scalar and summed.  &lt;br /&gt;
&lt;br /&gt;
Linear map: &lt;br /&gt;
&lt;br /&gt;
Little group (see stabilizer)&lt;br /&gt;
&lt;br /&gt;
Local actions&lt;br /&gt;
&lt;br /&gt;
Local hidden variable theory (see also hidden variable theory):&lt;br /&gt;
&lt;br /&gt;
Logical bit&lt;br /&gt;
&lt;br /&gt;
Matrix exponential&lt;br /&gt;
&lt;br /&gt;
Matrix properties&lt;br /&gt;
&lt;br /&gt;
Matrix transformation&lt;br /&gt;
&lt;br /&gt;
Measurement&lt;br /&gt;
&lt;br /&gt;
Minimum distance&lt;br /&gt;
&lt;br /&gt;
Modular arithmetic: When a number is divided into another and does not go evenly, there is left a remainder.  Modular arithmetic takes the information about what number was used to divide and what remainder is left to calculate how it will interact with another number.  For example, 13 can go into 5 twice with remainder 3, so its representation is 3mod 5, (pronounced &amp;quot;three modulo 5) which it has in common with any number satisfying 3+5x where x is an integer.  This usage of modulo has nothing to do with the physics usage of modulus.&lt;br /&gt;
&lt;br /&gt;
Modulus&lt;br /&gt;
&lt;br /&gt;
n,k,d code (see [n,k,d] code)&lt;br /&gt;
&lt;br /&gt;
No cloning theorem: There is no universal copying machine.  See [[Chapter 5 - Quantum Information: Basics and Simple Examples#No Cloning!|Section 5.2]].&lt;br /&gt;
&lt;br /&gt;
Noise&lt;br /&gt;
&lt;br /&gt;
Non-degenerate code&lt;br /&gt;
&lt;br /&gt;
Normalizer:&lt;br /&gt;
&lt;br /&gt;
Normalization: A process of scaling some set of numbers or functions in order that an operation including them returns a desired value.  For instance the set of all possible probabilities is usually scaled or normalized so they sum to one.&lt;br /&gt;
&lt;br /&gt;
One-to-one: A mapping where each domain element is mapped to exactly one range element, and each range element is mapped from one domain element.&lt;br /&gt;
&lt;br /&gt;
Onto: A mapping where each domain element mapped to at most one range element.&lt;br /&gt;
&lt;br /&gt;
Open system&lt;br /&gt;
&lt;br /&gt;
Operator&lt;br /&gt;
&lt;br /&gt;
Operator-sum representation (see SMR representation)&lt;br /&gt;
&lt;br /&gt;
Order of a group&lt;br /&gt;
&lt;br /&gt;
Ordered basis&lt;br /&gt;
&lt;br /&gt;
Orthogonal: Two vectors are orthogonal when their dot product is zero.&lt;br /&gt;
&lt;br /&gt;
Outer product&lt;br /&gt;
&lt;br /&gt;
P gate (not the phase gate):&lt;br /&gt;
&lt;br /&gt;
Parity&lt;br /&gt;
&lt;br /&gt;
Parity check (see inner product)&lt;br /&gt;
&lt;br /&gt;
Parity check matrix&lt;br /&gt;
&lt;br /&gt;
Partial trace&lt;br /&gt;
&lt;br /&gt;
Partition of a group&lt;br /&gt;
&lt;br /&gt;
Pauli group&lt;br /&gt;
&lt;br /&gt;
Pauli matrices: The X,Y,Z gates.&lt;br /&gt;
&lt;br /&gt;
Permutation:  &lt;br /&gt;
&lt;br /&gt;
Phase flip error&lt;br /&gt;
&lt;br /&gt;
Phase gate: See Z gate&lt;br /&gt;
&lt;br /&gt;
Planck's constant:&lt;br /&gt;
&lt;br /&gt;
Polarization&lt;br /&gt;
&lt;br /&gt;
Positive definite and semidefinite matrix:  A positive definite matrix is one whose eigenvalues are all greater than zero.  A positive semidefinite matrix has no negative eigenvalues.  &lt;br /&gt;
&lt;br /&gt;
Probability for existing in a state:&lt;br /&gt;
&lt;br /&gt;
Projector: A transformation such that &amp;lt;math&amp;gt;P^2=P&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
Projection postulate&lt;br /&gt;
&lt;br /&gt;
Pure state&lt;br /&gt;
&lt;br /&gt;
QKD: See quantum key distribution&lt;br /&gt;
&lt;br /&gt;
Quantum bit: See Qubit&lt;br /&gt;
&lt;br /&gt;
Quantum cryptography&lt;br /&gt;
&lt;br /&gt;
Quantum dense coding&lt;br /&gt;
&lt;br /&gt;
Quantum gate: A unitary transformation applied to one or more qubits.&lt;br /&gt;
&lt;br /&gt;
Quantum hamming bound&lt;br /&gt;
&lt;br /&gt;
Quantum key distribution:&lt;br /&gt;
&lt;br /&gt;
Quantum NOT gate: see X gate, but be careful, the NOT gate is only defined for qubits.&lt;br /&gt;
&lt;br /&gt;
Qubit: A Qubit is represented by two states of a quantum mechanical system. (1.3)&lt;br /&gt;
&lt;br /&gt;
Rank&lt;br /&gt;
&lt;br /&gt;
Rate of a code&lt;br /&gt;
&lt;br /&gt;
Reduced density operator&lt;br /&gt;
&lt;br /&gt;
Representation space&lt;br /&gt;
&lt;br /&gt;
Reversible  quantum operation:  An operation is reversible if for every state on which the operator can act, there exists an operation which restores the state to its original.&lt;br /&gt;
&lt;br /&gt;
RSA encryption&lt;br /&gt;
&lt;br /&gt;
Schmidt decomposition&lt;br /&gt;
&lt;br /&gt;
Schrodinger's Equation&lt;br /&gt;
&lt;br /&gt;
Set: Any mathematical construct.&lt;br /&gt;
&lt;br /&gt;
Shor's algorithm&lt;br /&gt;
&lt;br /&gt;
Shor's nine-bit quantum error correcting code&lt;br /&gt;
&lt;br /&gt;
Similarity transformation&lt;br /&gt;
&lt;br /&gt;
Singular values&lt;br /&gt;
&lt;br /&gt;
Singular value decomposition&lt;br /&gt;
&lt;br /&gt;
SMR representation&lt;br /&gt;
&lt;br /&gt;
Special unitary matrix&lt;br /&gt;
&lt;br /&gt;
Spin&lt;br /&gt;
&lt;br /&gt;
Spooky action at a distance&lt;br /&gt;
&lt;br /&gt;
Stabilizers of a group&lt;br /&gt;
&lt;br /&gt;
Stabilizer code&lt;br /&gt;
&lt;br /&gt;
Standard deviation&lt;br /&gt;
&lt;br /&gt;
Stationary subgroup (see stabilizer)&lt;br /&gt;
&lt;br /&gt;
Stirling's formula&lt;br /&gt;
&lt;br /&gt;
Subgroup&lt;br /&gt;
&lt;br /&gt;
Superposition: A qubit state in superposition, &amp;lt;math&amp;gt;\phi&amp;lt;/math&amp;gt; may be written as &amp;lt;math&amp;gt;|\phi&amp;gt;=\alpha|0&amp;gt;+\beta|1&amp;gt;&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;\alpha&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\beta&amp;lt;/math&amp;gt; are complex numbers.&lt;br /&gt;
&lt;br /&gt;
Syndrome measurement&lt;br /&gt;
&lt;br /&gt;
Taylor expansion&lt;br /&gt;
&lt;br /&gt;
Teleportation&lt;br /&gt;
&lt;br /&gt;
Tensor product&lt;br /&gt;
&lt;br /&gt;
Trace: The sum of the diagonal elements of a matrix.&lt;br /&gt;
&lt;br /&gt;
Transpose&lt;br /&gt;
&lt;br /&gt;
Trivial representation&lt;br /&gt;
&lt;br /&gt;
Turing machine&lt;br /&gt;
&lt;br /&gt;
Uncertainty principle&lt;br /&gt;
&lt;br /&gt;
Unitary matrix&lt;br /&gt;
&lt;br /&gt;
Unitary transformation: A transformation which leaves the magnitude of any object it transforms the same.&lt;br /&gt;
&lt;br /&gt;
Universal quantum computing&lt;br /&gt;
&lt;br /&gt;
Universal set of gates (universality) (2.6)&lt;br /&gt;
&lt;br /&gt;
Variance&lt;br /&gt;
&lt;br /&gt;
Vector: A directed quantity.&lt;br /&gt;
&lt;br /&gt;
Vector space&lt;br /&gt;
&lt;br /&gt;
Weight of a vector (see Hamming weight)&lt;br /&gt;
&lt;br /&gt;
Weight of an operator: The number of non-identity elements in the tensor product.&lt;br /&gt;
&lt;br /&gt;
Wigner-Clebsch-Gordon Coefficients&lt;br /&gt;
&lt;br /&gt;
X gate (2.3.2)&lt;br /&gt;
&lt;br /&gt;
Y gate&lt;br /&gt;
&lt;br /&gt;
Z gate, or phase-flip gate (2.3.2)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&amp;diff=2355</id>
		<title>Appendix C - Vectors and Linear Algebra</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&amp;diff=2355"/>
		<updated>2013-09-17T02:08:58Z</updated>

		<summary type="html">&lt;p&gt;Hilary: Inner and Outer products after Hermitian&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;===Introduction===&lt;br /&gt;
&lt;br /&gt;
This appendix introduces some aspects of linear algebra and complex&lt;br /&gt;
algebra that will be helpful for the course.  In addition, Dirac&lt;br /&gt;
notation is introduced and explained.&lt;br /&gt;
&lt;br /&gt;
===Vectors===&lt;br /&gt;
&lt;br /&gt;
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics.  &lt;br /&gt;
&lt;br /&gt;
====Real Vectors====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The simple definition of a vector --- an object that has magnitude and&lt;br /&gt;
direction --- is helpful to keep in mind even when dealing with complex&lt;br /&gt;
and/or abstract vectors as we will here.  In three dimensional space,&lt;br /&gt;
a vector is often written as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where the hat (&amp;lt;math&amp;gt;\hat{\cdot}\,\!&amp;lt;/math&amp;gt;) denotes a unit vector and the components&lt;br /&gt;
&amp;lt;math&amp;gt;v_i\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;i = x,y,z\,\!&amp;lt;/math&amp;gt; are just numbers.  The unit vectors are also&lt;br /&gt;
known as ''basis'' vectors. &amp;lt;!-- \index{basis vectors!real} --&amp;gt; &lt;br /&gt;
This is because any vector&lt;br /&gt;
in real three-dimensional space can be written in terms of these unit/basis vectors.  In&lt;br /&gt;
some sense they are the basic components of any vector.  Other basis&lt;br /&gt;
vectors could be used, however, such as in spherical and cylindrical coordinates.  When dealing with more abstract and/or complex vectors,&lt;br /&gt;
it is often helpful to ask what one would do for an ordinary&lt;br /&gt;
three-dimensional vector.  For example, properties of unit vectors,&lt;br /&gt;
dot products, etc. in three-dimensions are similar to the analogous&lt;br /&gt;
constructions in higher dimensions.  &lt;br /&gt;
&lt;br /&gt;
The ''inner product'',&amp;lt;!-- \index{inner product}--&amp;gt; or ''dot product'',&amp;lt;!--\index{dot product}--&amp;gt; for two real three-dimensional vectors,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; &lt;br /&gt;
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
can be computed as follows:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
For the inner product of &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt; with itself, we get the square of&lt;br /&gt;
the magnitude of &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt;, denoted &amp;lt;math&amp;gt;|\vec{v}|^2\,\!&amp;lt;/math&amp;gt;:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +&lt;br /&gt;
v_zv_z=v_x^2+v_y^2+v_z^2. &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If we want a unit vector in the direction of &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt;, we can simply divide it&lt;br /&gt;
by its magnitude:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\hat{v} = \frac{\vec{v}}{|\vec{v}|}.  &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Now, of course, &amp;lt;math&amp;gt;\hat{v}\cdot\hat{v}= 1\,\!&amp;lt;/math&amp;gt;, which can easily be checked.  &lt;br /&gt;
&lt;br /&gt;
There are several ways to represent a vector.  The ones we will use&lt;br /&gt;
most often are column and row vector notations.  So, for example, we&lt;br /&gt;
could write the vector above as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z&lt;br /&gt;
  \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
In this case, our unit vectors are represented by the following: &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0&lt;br /&gt;
  \end{array}\right), \;\;  &lt;br /&gt;
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0&lt;br /&gt;
  \end{array}\right), \;\;&lt;br /&gt;
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
We next turn to the subject of complex vectors and the relevant&lt;br /&gt;
notation. &lt;br /&gt;
We will see how to compute the inner product later, since some other&lt;br /&gt;
definitions are required.&lt;br /&gt;
&lt;br /&gt;
====Complex Vectors====&lt;br /&gt;
&lt;br /&gt;
For complex vectors in quantum mechanics, Dirac notation is used most often.  This notation uses a &amp;lt;math&amp;gt;\left\vert \cdot \right\rangle\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
called a ''ket'', for a vector.  So our vector &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt; would be&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert v \right\rangle  = \left(\begin{array}{c} v_x \\ v_y \\ v_z&lt;br /&gt;
  \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For qubits, i.e. two-state quantum systems, complex vectors will often be used:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align} \left\vert \psi \right\rangle &amp;amp;= \left(\begin{array}{c} \alpha \\ \beta &lt;br /&gt;
  \end{array}\right) \\&lt;br /&gt;
           &amp;amp;=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}&amp;lt;/math&amp;gt;|C.1}} &lt;br /&gt;
where&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 &lt;br /&gt;
  \end{array}\right), \;\;\mbox{and} \;\;&lt;br /&gt;
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 &lt;br /&gt;
  \end{array}\right)&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
are the basis vectors.  The two numbers &amp;lt;math&amp;gt;\alpha\,\!&amp;lt;/math&amp;gt; and&lt;br /&gt;
&amp;lt;math&amp;gt;\beta\,\!&amp;lt;/math&amp;gt; are complex numbers, so the vector is said to&lt;br /&gt;
be a complex vector.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
===Linear Algebra: Matrices===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are many aspects of linear algebra that are quite useful in&lt;br /&gt;
quantum mechanics.  We will briefly discuss several of these aspects here.&lt;br /&gt;
First, some definitions and properties are provided that will&lt;br /&gt;
be useful.  Some familiarity with matrices&lt;br /&gt;
will be assumed, although many basic definitions are also included.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Let us denote some &amp;lt;math&amp;gt;m\times n\,\!&amp;lt;/math&amp;gt; matrix by &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt;.  The set of all &amp;lt;math&amp;gt;m\times&lt;br /&gt;
n\,\!&amp;lt;/math&amp;gt; matrices with real entries is &amp;lt;math&amp;gt;M(n\times m,\mathbb{R})\,\!&amp;lt;/math&amp;gt;.  Such matrices&lt;br /&gt;
are said to be real since they have all real entries.  Similarly, the&lt;br /&gt;
set of &amp;lt;math&amp;gt;m\times n\,\!&amp;lt;/math&amp;gt; complex matrices is &amp;lt;math&amp;gt;M(m\times n,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.  For the&lt;br /&gt;
set of square &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; complex matrices, we simply write&lt;br /&gt;
&amp;lt;math&amp;gt;M(n,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
We will also refer to the set of matrix elements, &amp;lt;math&amp;gt;a_{ij}\,\!&amp;lt;/math&amp;gt;, where the&lt;br /&gt;
first index (&amp;lt;math&amp;gt;i\,\!&amp;lt;/math&amp;gt; in this case) labels the row and the second &amp;lt;math&amp;gt;(j)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
labels the column.  Thus the element &amp;lt;math&amp;gt;a_{23}\,\!&amp;lt;/math&amp;gt; is the element in the&lt;br /&gt;
second row and third column.  A comma is inserted if there is some&lt;br /&gt;
ambiguity.  For example, in a large matrix the element in the&lt;br /&gt;
2nd row and 12th&lt;br /&gt;
column is written as &amp;lt;math&amp;gt;a_{2,12}\,\!&amp;lt;/math&amp;gt; to distinguish between the&lt;br /&gt;
21st row and 2nd column.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Complex Conjugate====&lt;br /&gt;
&lt;br /&gt;
The ''complex conjugate of a matrix'' &amp;lt;!-- \index{complex conjugate!of a matrix}--&amp;gt;&lt;br /&gt;
is the matrix with each element replaced by its complex conjugate.  In&lt;br /&gt;
other words, to take the complex conjugate of a matrix, one takes the&lt;br /&gt;
complex conjugate of each entry in the matrix.  We denote the complex&lt;br /&gt;
conjugate with a star, like this: &amp;lt;math&amp;gt;A^*\,\!&amp;lt;/math&amp;gt;.  For example,&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
A^* &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11} &amp;amp; a_{12} &amp;amp; a_{13} \\&lt;br /&gt;
        a_{21} &amp;amp; a_{22} &amp;amp; a_{23} \\&lt;br /&gt;
        a_{31} &amp;amp; a_{32} &amp;amp; a_{33} \end{array}\right)^*  \\&lt;br /&gt;
    &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11}^* &amp;amp; a_{12}^* &amp;amp; a_{13}^* \\&lt;br /&gt;
        a_{21}^* &amp;amp; a_{22}^* &amp;amp; a_{23}^* \\&lt;br /&gt;
        a_{31}^* &amp;amp; a_{32}^* &amp;amp; a_{33}^* \end{array}\right). &lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.2}}&lt;br /&gt;
(Notice that the notation for a matrix is a capital letter, whereas&lt;br /&gt;
the entries are represented by lower case&lt;br /&gt;
letters.)&lt;br /&gt;
&lt;br /&gt;
====Transpose====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The ''transpose'' &amp;lt;!-- \index{transpose} --&amp;gt; of a matrix is the same set of&lt;br /&gt;
elements, but now the first row becomes the first column, the second row&lt;br /&gt;
becomes the second column, and so on.  Thus the rows and columns are&lt;br /&gt;
interchanged.  For example, for a square &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; matrix, the&lt;br /&gt;
transpose is given by&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
A^T &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11} &amp;amp; a_{12} &amp;amp; a_{13} \\&lt;br /&gt;
        a_{21} &amp;amp; a_{22} &amp;amp; a_{23} \\&lt;br /&gt;
        a_{31} &amp;amp; a_{32} &amp;amp; a_{33} \end{array}\right)^T \\&lt;br /&gt;
    &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11} &amp;amp; a_{21} &amp;amp; a_{31} \\&lt;br /&gt;
        a_{12} &amp;amp; a_{22} &amp;amp; a_{32} \\&lt;br /&gt;
        a_{13} &amp;amp; a_{23} &amp;amp; a_{33} \end{array}\right). &lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.3}}&lt;br /&gt;
&lt;br /&gt;
====Hermitian Conjugate====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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,&lt;br /&gt;
(&amp;lt;math&amp;gt;\dagger\,\!&amp;lt;/math&amp;gt;):&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
(A^T)^* = (A^*)^T \equiv A^\dagger.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.4}}&lt;br /&gt;
For our &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; example, &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
A^\dagger = \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11}^* &amp;amp; a_{21}^* &amp;amp; a_{31}^* \\&lt;br /&gt;
        a_{12}^* &amp;amp; a_{22}^* &amp;amp; a_{32}^* \\&lt;br /&gt;
        a_{13}^* &amp;amp; a_{23}^* &amp;amp; a_{33}^* \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If a matrix is its own Hermitian conjugate, i.e. &amp;lt;math&amp;gt;A^\dagger = A\,\!&amp;lt;/math&amp;gt;, then&lt;br /&gt;
we call it a ''Hermitian matrix''.  &amp;lt;!-- \index{Hermitian matrix}--&amp;gt;&lt;br /&gt;
(Clearly this is only possible for square matrices.) Hermitian&lt;br /&gt;
matrices are very important in quantum mechanics since their&lt;br /&gt;
eigenvalues are real.  (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)&lt;br /&gt;
&lt;br /&gt;
====Index Notation====&lt;br /&gt;
&lt;br /&gt;
Very often we write the product of two matrices &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt; simply as&lt;br /&gt;
&amp;lt;math&amp;gt;AB\,\!&amp;lt;/math&amp;gt; and let &amp;lt;math&amp;gt;C=AB\,\!&amp;lt;/math&amp;gt;.  However, it is also quite useful to write this&lt;br /&gt;
in component form.  In this case, if these are &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrices, the component form will be &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This says that the element in the &amp;lt;math&amp;gt;i^{\mbox{th}}\,\!&amp;lt;/math&amp;gt; row and&lt;br /&gt;
&amp;lt;math&amp;gt;j^{\mbox{th}}\,\!&amp;lt;/math&amp;gt; column of the matrix &amp;lt;math&amp;gt;C\,\!&amp;lt;/math&amp;gt; is the sum &amp;lt;math&amp;gt;\sum_{j=1}^n&lt;br /&gt;
a_{ij}b_{jk}\,\!&amp;lt;/math&amp;gt;.  The transpose of &amp;lt;math&amp;gt;C\,\!&amp;lt;/math&amp;gt; has elements&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Now if we were to transpose &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt; as well, this would read&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_{j=1}^n b^T_{ij} a^T_{jk}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This gives us a way of seeing the general rule that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
C^T = B^TA^T.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
It follows that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
C^\dagger = B^\dagger A^\dagger.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====The Trace====&lt;br /&gt;
&lt;br /&gt;
The ''trace'' &amp;lt;!-- \index{trace}--&amp;gt; of a matrix is the sum of the diagonal&lt;br /&gt;
elements and is denoted &amp;lt;math&amp;gt;\mbox{Tr}\,\!&amp;lt;/math&amp;gt;.  So for example, the trace of an&lt;br /&gt;
&amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some useful properties of the trace are the following:&lt;br /&gt;
#&amp;lt;math&amp;gt;\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
Using the first of these results,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This relation is used so often that we state it here explicitly.&lt;br /&gt;
&lt;br /&gt;
====The Determinant====&lt;br /&gt;
&lt;br /&gt;
For a square matrix, the determinant is quite a useful thing.  For&lt;br /&gt;
example, an &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrix is invertible if and only if its&lt;br /&gt;
determinant is not zero.  So let us define the determinant and give&lt;br /&gt;
some properties and examples.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The ''determinant''&amp;lt;!--\index{determinant}--&amp;gt; of a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; matrix, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
N = \left(\begin{array}{cc}&lt;br /&gt;
                 a &amp;amp; b \\&lt;br /&gt;
                 c &amp;amp; d \end{array}\right),&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.5}}&lt;br /&gt;
is given by&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(N) = ad-bc.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.6}}&lt;br /&gt;
Higher-order determinants can be written in terms of smaller ones in a recursive way.  For example, let &lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
M = \left(\begin{array}{ccc}&lt;br /&gt;
                 m_{11} &amp;amp; m_{12} &amp;amp; m_{13} \\&lt;br /&gt;
                 m_{21} &amp;amp; m_{22} &amp;amp; m_{23} \\&lt;br /&gt;
                 m_{31} &amp;amp; m_{32} &amp;amp; m_{33}\end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;  &lt;br /&gt;
Then &lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\det(M) = m_{11}\det\left(\begin{array}{cc}&lt;br /&gt;
                 m_{21} &amp;amp; m_{22} \\&lt;br /&gt;
                 m_{31} &amp;amp; m_{32} \end{array}\right) &lt;br /&gt;
- m_{12}\det\left(\begin{array}{cc}&lt;br /&gt;
                 m_{21} &amp;amp; m_{23} \\&lt;br /&gt;
                 m_{31} &amp;amp; m_{33} \end{array}\right)&lt;br /&gt;
+m_{13}\det\left(\begin{array}{cc}&lt;br /&gt;
                 m_{22} &amp;amp; m_{23} \\&lt;br /&gt;
                 m_{32} &amp;amp; m_{33} \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The ''determinant''&amp;lt;!-- \index{determinant}--&amp;gt; of a matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; can be&lt;br /&gt;
also be written in terms of its components as &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}&lt;br /&gt;
a_{1i}a_{2j}a_{3k}a_{4l} ...,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.7}}&lt;br /&gt;
where the symbol &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijkl...} = \begin{cases}&lt;br /&gt;
                       +1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\&lt;br /&gt;
                       -1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\&lt;br /&gt;
                       \;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).&lt;br /&gt;
                      \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.8}}&lt;br /&gt;
&lt;br /&gt;
Let us consider the example of the &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; given&lt;br /&gt;
above.  The determinant can be calculated by&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(A) = \sum_{i,j,k} \epsilon_{ijk}&lt;br /&gt;
a_{1i}a_{2j}a_{3k},&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where, explicitly, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}),&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.9}}&lt;br /&gt;
so that &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\det(A) &amp;amp;=&amp;amp; \epsilon_{123}a_{11}a_{22}a_{33} &lt;br /&gt;
         +\epsilon_{132}a_{11}a_{23}a_{32}&lt;br /&gt;
         +\epsilon_{231}a_{12}a_{23}a_{31}  \\&lt;br /&gt;
       &amp;amp;&amp;amp;+\epsilon_{213}a_{12}a_{21}a_{33}&lt;br /&gt;
         +\epsilon_{312}a_{13}a_{21}a_{32}&lt;br /&gt;
         +\epsilon_{321}a_{13}a_{22}a_{31}.&lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.10}}&lt;br /&gt;
Now given the values of &amp;lt;math&amp;gt;\epsilon_{ijk}\,\!&amp;lt;/math&amp;gt; in [[#eqC.9|Eq. C.9]],&lt;br /&gt;
this is&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} &lt;br /&gt;
         - a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The determinant has several properties that are useful to know.  A few are listed here:  &lt;br /&gt;
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: &amp;lt;center&amp;gt;&amp;lt;math&amp;gt; \det(A) = \det(A^T).\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
#The determinant of a product is the product of determinants:    &amp;lt;center&amp;gt;&amp;lt;math&amp;gt; \det(AB) = \det(A)\det(B).\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
From this last property, another specific property can be derived.&lt;br /&gt;
If we take the determinant of the product of a matrix and its&lt;br /&gt;
inverse, we find&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
since the determinant of the identity is one.  This implies that&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(U^{-1}) = \frac{1}{\det(U)}.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====The Inverse of a Matrix====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The inverse &amp;lt;!-- \index{inverse}--&amp;gt; of a square matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; is another matrix,&lt;br /&gt;
denoted &amp;lt;math&amp;gt;A^{-1}\,\!&amp;lt;/math&amp;gt;, such that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
AA^{-1} = A^{-1}A = \mathbb{I},&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt;\mathbb{I}\,\!&amp;lt;/math&amp;gt; is the identity matrix consisting of zeroes everywhere&lt;br /&gt;
except the diagonal, which has ones.  For example, the &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
identity matrix is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 1 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 1&lt;br /&gt;
  \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.''  Thus one only needs to calculate the&lt;br /&gt;
determinant to see if a matrix has an inverse or not.&lt;br /&gt;
&lt;br /&gt;
====Hermitian Matrices====&lt;br /&gt;
&lt;br /&gt;
Hermitian matrices are important for a variety of reasons; primarily, it is because their eigenvalues are real.  Thus Hermitian matrices are used to represent density operators and density matrices, as well as Hamiltonians.  The density operator is a positive semi-definite Hermitian matrix (it has no negative eigenvalues) that has its trace equal to one.  In any case, it is often desirable to represent &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrices using a real linear combination of a complete set of &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrices.  A set of &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrices is complete if any Hermitian matrix can be represented in terms of the set.  Let &amp;lt;math&amp;gt;\{\lambda_i\}\,\!&amp;lt;/math&amp;gt; be a complete set.  Then any Hermitian matrix can be represented by &amp;lt;math&amp;gt;\sum_i a_i \lambda_i\,\!&amp;lt;/math&amp;gt;.  The set can always be taken to be a set of traceless Hermitian matrices and the identity matrix.  This is convenient for the density matrix (its trace is one) because the identity part of an &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrix is &amp;lt;math&amp;gt;(1/N)\mathbb{I}\,\!&amp;lt;/math&amp;gt; if we take all others in the set to be traceless.  For the Hamiltonian, the set consists of a traceless part and an identity part where identity part just gives an overall phase which can often be neglected.  &lt;br /&gt;
&lt;br /&gt;
One example of such a set which is extremely useful is the set of Pauli matrices.  These are discussed in detail in [[Chapter 2 - Qubits and Collections of Qubits|Chapter 2]] and in particular in [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|Section 2.4]].&lt;br /&gt;
&lt;br /&gt;
====Unitary Matrices====&lt;br /&gt;
&lt;br /&gt;
A ''unitary matrix'' &amp;lt;!-- \index{unitary matrix} --&amp;gt; &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; is one whose&lt;br /&gt;
inverse is also its Hermitian conjugate, &amp;lt;math&amp;gt;U^\dagger = U^{-1}\,\!&amp;lt;/math&amp;gt;, so that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
U^\dagger U = UU^\dagger = \mathbb{I}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.&amp;lt;!-- \index{special unitary matrix}--&amp;gt;  The set of&lt;br /&gt;
&amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; unitary matrices is denoted&lt;br /&gt;
&amp;lt;math&amp;gt;U(n)\,\!&amp;lt;/math&amp;gt; and the set of special unitary matrices is denoted &amp;lt;math&amp;gt;SU(n)\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
Unitary matrices are particularly important in quantum mechanics&lt;br /&gt;
because they describe the evolution of quantum states.&lt;br /&gt;
They have this ability due to the fact that the rows and columns of unitary matrices (viewed as vectors) are orthonormal. (This is made clear in an example below.)  This means that when&lt;br /&gt;
they act on a basis vector of the form&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt; \left\vert j\right\rangle = &lt;br /&gt;
 \left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 &lt;br /&gt;
  \end{array}\right), &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.11}}&lt;br /&gt;
with a single 1, in say the &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;th spot, and zeroes everywhere else, the result is a normalized complex vector.  Acting on a set of&lt;br /&gt;
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]]&lt;br /&gt;
will produce another orthonormal set.  &lt;br /&gt;
&lt;br /&gt;
Let us consider the example of a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; unitary matrix, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
U = \left(\begin{array}{cc} &lt;br /&gt;
              a &amp;amp; b \\ &lt;br /&gt;
              c &amp;amp; d &lt;br /&gt;
           \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.12}}&lt;br /&gt;
The inverse of this matrix is the Hermitian conjugate, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
U ^{-1} = U^\dagger = \left(\begin{array}{cc} &lt;br /&gt;
                         a^* &amp;amp; c^* \\ &lt;br /&gt;
                         b^* &amp;amp; d^* &lt;br /&gt;
                       \end{array}\right),&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.13}}&lt;br /&gt;
provided that the matrix &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; satisfies the constraints&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
|a|^2 + |b|^2 = 1, \; &amp;amp; \; ac^*+bd^* =0  \\&lt;br /&gt;
ca^*+db^*=0,  \;      &amp;amp;  \; |c|^2 + |d|^2 =1,&lt;br /&gt;
\end{align}\,\!&amp;lt;/math&amp;gt;|C.14}}&lt;br /&gt;
and&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
|a|^2 + |c|^2 = 1, \; &amp;amp; \; ba^*+dc^* =0  \\&lt;br /&gt;
b^*a+d^*c=0,  \;      &amp;amp;  \; |b|^2 + |d|^2 =1.&lt;br /&gt;
\end{align}\,\!&amp;lt;/math&amp;gt;|C.15}}&lt;br /&gt;
Looking at each row as a vector, the constraints in&lt;br /&gt;
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the&lt;br /&gt;
vectors forming the rows.  Similarly, the constraints in&lt;br /&gt;
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the&lt;br /&gt;
vectors forming the columns.&lt;br /&gt;
&lt;br /&gt;
====Inner and Outer Products====&lt;br /&gt;
&lt;br /&gt;
Now that we have a definition for the Hermitian conjugate, we consider the&lt;br /&gt;
case for a &amp;lt;math&amp;gt;1\times n\,\!&amp;lt;/math&amp;gt; matrix, i.e. a vector.  In Dirac notation, this is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The Hermitian conjugate comes up so often that we use the following&lt;br /&gt;
notation for vectors:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\&lt;br /&gt;
    \beta \end{array}\right)^\dagger &lt;br /&gt;
 = \left( \alpha^*, \; \beta^* \right).  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This is a row vector and in Dirac notation is denoted by the symbol &amp;lt;math&amp;gt;\left\langle\cdot \right\vert\!&amp;lt;/math&amp;gt;, which is called a ''bra''.  Let us consider a second complex vector, &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta&lt;br /&gt;
  \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The ''inner product'' &amp;lt;!-- \index{inner product}--&amp;gt; between &amp;lt;math&amp;gt;\left\vert\psi\right\rangle\,\!&amp;lt;/math&amp;gt; and &lt;br /&gt;
&amp;lt;math&amp;gt;\left\vert\phi\right\rangle\,\!&amp;lt;/math&amp;gt; is computed as follows:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
 \left\langle\phi\mid\psi\right\rangle &amp;amp; \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle   \\&lt;br /&gt;
                  &amp;amp;= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right)   \\&lt;br /&gt;
                  &amp;amp;= \gamma^*\alpha + \delta^*\beta.&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.16}}&lt;br /&gt;
&lt;br /&gt;
The ''outer product'' between these same two vectors is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt; &lt;br /&gt;
\begin{align} (\left\vert \phi \right\rangle )(\left\vert \psi \right\rangle)^\dagger &lt;br /&gt;
 &amp;amp;=  \left\vert \phi \right\rangle \left\langle \psi \right\vert \\&lt;br /&gt;
&amp;amp;= \left(\begin{array}{c} \gamma \\ \delta \end{array}\right)&lt;br /&gt;
\left(\begin{array}{c} \alpha \\ \beta   \end{array}\right)^\dagger \\&lt;br /&gt;
           &amp;amp;= \left(\begin{array}{c} \gamma \\ \delta \end{array}\right) \left(\begin{array}{cc} \alpha^* &amp;amp; \beta^*   \end{array}\right) \\  &lt;br /&gt;
           &amp;amp;=   \left(\begin{array}{cc} \gamma\alpha^* &amp;amp; \gamma\beta^* \\  \delta\alpha^* &amp;amp; \delta\beta^*  \end{array}\right) \end{align}\;\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===More Dirac Notation===&lt;br /&gt;
&lt;br /&gt;
If these two vectors are ''orthogonal'', &amp;lt;!-- \index{orthogonal!vectors} --&amp;gt;&lt;br /&gt;
then their inner product is zero, or &amp;lt;math&amp;gt;\left\langle\phi\mid\psi\right\rangle =0\,\!&amp;lt;/math&amp;gt;.  (The &amp;lt;math&amp;gt; \left\langle\phi\mid\psi\right\rangle \,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
is called a ''bracket'', which is the product of the ''bra'' and the ''ket''.)  The inner product of &amp;lt;math&amp;gt;\left\vert\psi\right\rangle\,\!&amp;lt;/math&amp;gt; with itself is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This vector is considered normalized when &amp;lt;math&amp;gt;\left\langle\psi\mid\psi\right\rangle = 1\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
More generally, we will consider vectors in &amp;lt;math&amp;gt;N\,\!&amp;lt;/math&amp;gt; dimensions.  In this&lt;br /&gt;
case we write the vector in terms of a set of basis vectors,&lt;br /&gt;
&amp;lt;math&amp;gt;\{\left\vert i\right\rangle\}\,\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;i = 0,1,2,...N-1\,\!&amp;lt;/math&amp;gt;.  This is an ordered set of&lt;br /&gt;
vectors which are labeled simply by integers.  If the set is orthogonal,&lt;br /&gt;
then &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If they are normalized, then &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If both of these are true, i.e. the entire set is orthonormal, we can&lt;br /&gt;
write&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i\mid j\right\rangle = \delta_{ij}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where the symbol &amp;lt;math&amp;gt;\delta_{ij}\,\!&amp;lt;/math&amp;gt; is called the Kronecker delta &amp;lt;!-- \index{Kronecker delta} --&amp;gt; and is defined by &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\delta_{ij} = \begin{cases}&lt;br /&gt;
               1, &amp;amp; \mbox{if } i=j, \\&lt;br /&gt;
               0, &amp;amp; \mbox{if } i\neq j.&lt;br /&gt;
              \end{cases}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.17}}&lt;br /&gt;
Now consider &amp;lt;math&amp;gt;(N+1)\,\!&amp;lt;/math&amp;gt;-dimensional vectors by letting two such vectors&lt;br /&gt;
be expressed in the same basis as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
and&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Then the inner product &amp;lt;!--\index{inner product}--&amp;gt; is&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\langle\Psi\mid\Phi\right\rangle &amp;amp;= \left(\sum_{i=0}^{N}&lt;br /&gt;
             \alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right)  \\&lt;br /&gt;
                 &amp;amp;= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle  \\&lt;br /&gt;
                 &amp;amp;= \sum_{ij} \alpha_i^*\beta_j\delta_{ij}  \\&lt;br /&gt;
                 &amp;amp;= \sum_i\alpha^*_i\beta_i,&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.18}}&lt;br /&gt;
where the fact that the delta function is zero unless&lt;br /&gt;
&amp;lt;math&amp;gt;i=j\,\!&amp;lt;/math&amp;gt; is used to obtain the last equality.  Taking the inner product of a vector&lt;br /&gt;
with itself will get&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This immediately gives us a very important property of the inner&lt;br /&gt;
product.  It tells us that, in general,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0&lt;br /&gt;
\Leftrightarrow \left\vert\Phi\right\rangle = 0. &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
(The symbol &amp;lt;math&amp;gt;\Leftrightarrow\,\!&amp;lt;/math&amp;gt; means &amp;lt;nowiki&amp;gt;&amp;quot;if and only if,&amp;quot;&amp;lt;/nowiki&amp;gt; sometimes written as &amp;lt;nowiki&amp;gt;&amp;quot;iff.&amp;quot;&amp;lt;/nowiki&amp;gt;)  &lt;br /&gt;
&lt;br /&gt;
We could also expand a vector in a different basis.  Let us suppose&lt;br /&gt;
that the set &amp;lt;math&amp;gt;\{\left\vert e_k \right\rangle\}\,\!&amp;lt;/math&amp;gt; is an orthonormal basis &amp;lt;math&amp;gt;(\left\langle e_k \mid e_l\right\rangle =&lt;br /&gt;
\delta_{kl})\,\!&amp;lt;/math&amp;gt; that is different from the one considered earlier.  We&lt;br /&gt;
could expand our vector &amp;lt;math&amp;gt;\left\vert\Psi\right\rangle\,\!&amp;lt;/math&amp;gt; in terms of our new basis by&lt;br /&gt;
expanding our new basis in terms of our old basis.  Let us first&lt;br /&gt;
expand the &amp;lt;math&amp;gt;\left\vert e_k\right\rangle\,\!&amp;lt;/math&amp;gt; in terms of the &amp;lt;math&amp;gt;\left\vert j\right\rangle\,\!&amp;lt;/math&amp;gt;:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.19}}&lt;br /&gt;
so that &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert \Psi\right\rangle &amp;amp;= \sum_j \alpha_j\left\vert j\right\rangle  \\&lt;br /&gt;
           &amp;amp;= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle  \\ &lt;br /&gt;
           &amp;amp;= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.20}}&lt;br /&gt;
where &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.21}}&lt;br /&gt;
Notice that the insertion of &amp;lt;math&amp;gt;\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!&amp;lt;/math&amp;gt; didn't do anything to our original vector; it is the same vector, just in a&lt;br /&gt;
different basis.  Therefore, this is effectively the identity operator,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This is an important and quite useful relation.  &lt;br /&gt;
To interpret Eq.[[#eqC.19|(C.19)]], we can draw a close&lt;br /&gt;
analogy with three-dimensional real vectors.  The inner product&lt;br /&gt;
&amp;lt;math&amp;gt;\left\vert e_k \right\rangle \left\langle j \right\vert\,\!&amp;lt;/math&amp;gt; can be interpreted as the projection of one vector onto&lt;br /&gt;
another.  This provides the part of &amp;lt;math&amp;gt;\left\vert j \right\rangle\,\!&amp;lt;/math&amp;gt; along &amp;lt;math&amp;gt;\left\vert e_k \right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
===Transformations===&lt;br /&gt;
&lt;br /&gt;
Suppose we have two different orthogonal bases, &amp;lt;math&amp;gt;\{e_k\}\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;\{j\}\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
The numbers &amp;lt;math&amp;gt;\left\langle e_k\mid j\right\rangle\,\!&amp;lt;/math&amp;gt; for all the different &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;j\,\!&amp;lt;/math&amp;gt; are&lt;br /&gt;
often referred to as matrix elements since the set forms a matrix, with&lt;br /&gt;
&amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt; labelling the rows and &amp;lt;math&amp;gt;j\,\!&amp;lt;/math&amp;gt; labelling the columns.  Thus we&lt;br /&gt;
can represent the transformation from one basis to another with a matrix&lt;br /&gt;
transformation.  Let &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; be the matrix with elements &amp;lt;math&amp;gt;m_{kj} =&lt;br /&gt;
\left\langle e_k\mid j\right\rangle\,\!&amp;lt;/math&amp;gt;.  The transformation from one basis to another,&lt;br /&gt;
written in terms of the coefficients of &amp;lt;math&amp;gt;\left\vert\Psi\right\rangle\,\!&amp;lt;/math&amp;gt;, is &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt; A^\prime = MA, &amp;lt;/math&amp;gt;|C.22}}&lt;br /&gt;
where &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\&lt;br /&gt;
    \alpha_n^\prime \end{array}\right), \;\; &lt;br /&gt;
\mbox{ and } \;\;&lt;br /&gt;
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\&lt;br /&gt;
    \alpha_n\end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This sort of transformation is a change of basis.  Oftentimes when one vector is transformed to another the transformation can be viewed as a transformation of the components of the vector and is also represented by a matrix.  Thus transformations can either be&lt;br /&gt;
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.23}}&lt;br /&gt;
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.  &lt;br /&gt;
&lt;br /&gt;
For a general transformation matrix &amp;lt;math&amp;gt;T\,\!&amp;lt;/math&amp;gt; acting on a vector,&lt;br /&gt;
the matrix elements in a particular basis &amp;lt;math&amp;gt;\left\vert i\right\rangle\,\!&amp;lt;/math&amp;gt; are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
just as elements of a vector can be found using&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i\mid \Psi \right\rangle = \alpha_i.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Transformations of a Qubit====&lt;br /&gt;
&lt;br /&gt;
It is worth belaboring the point somewhat and presenting several ways in which to parametrize the set of transformations of a qubit.  A qubit state is represented by a complex two-dimensional vector that has been normalized to one:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert \psi\right\rangle = \alpha_0 \left\vert 0 \right\rangle + \alpha_1 \left\vert 1\right\rangle = \left(\begin{array}{c} \alpha_0 \\ \alpha_1 \end{array}\right), \;\;\;\; |\alpha_0|^2 + |\alpha_1|^2 = 1.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The most general matrix transformation that will take this to any other state of the same form (complex, 2-d vector with unit norm) is a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; unitary matrix.  In [[Chapter 2 - Qubits and Collections of Qubits|Chapter 2]], several specific examples of qubit transformations were given; in [[Chapter 3 - Physics of Quantum Information|Chapter 3]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|Section 3.4]] it was stated that an element of SU(2) can be written as (see [[Chapter 3 - Physics of Quantum Information#Exponentian of a Matrix|Section 3.2.1, Exponentiation of a Matrix]], in particular [[Chapter 3 - Physics of Quantum Information#eq3.8|Eq. (3.8)]])&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
U(\theta) &amp;amp;= \exp(-i\vec{n}\cdot\vec{\sigma} \theta/2) \\&lt;br /&gt;
          &amp;amp;= (\mathbb{I}\cos(\theta/2) -i\vec{n}\cdot\vec{\sigma} \sin(\theta/2))&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.24}}&lt;br /&gt;
where &amp;lt;math&amp;gt;\vec{n}\,\!&amp;lt;/math&amp;gt; is a unit vector, &amp;lt;math&amp;gt;|\vec{n}|=1\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;\vec{n}\cdot\vec{\sigma} =&lt;br /&gt;
n_1\sigma_1+n_2\sigma_2+n_3\sigma_3\,\!&amp;lt;/math&amp;gt;. &lt;br /&gt;
Explicitly, this is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
 \exp(-i\vec{n}\cdot\vec{\sigma} \theta/2) &amp;amp;= \left(\begin{array}{cc}&lt;br /&gt;
                                  1 &amp;amp; 0 \\ &lt;br /&gt;
                                  0 &amp;amp; 1 \end{array}\right)\cos(\theta/2) \\&lt;br /&gt;
                        &amp;amp; \;\;\;   + (-i)\left[ n_1\left(\begin{array}{cc}&lt;br /&gt;
                                  0 &amp;amp; 1 \\ &lt;br /&gt;
                                  1 &amp;amp; 0 \end{array}\right)&lt;br /&gt;
                              + n_2\left(\begin{array}{cc}&lt;br /&gt;
                                  0 &amp;amp; -i \\ &lt;br /&gt;
                                  i &amp;amp; 0 \end{array}\right)&lt;br /&gt;
                              + n_3\left(\begin{array}{cc}&lt;br /&gt;
                                  1 &amp;amp; 0 \\ &lt;br /&gt;
                                  0 &amp;amp; -1 \end{array}\right)\right]\sin(\theta/2) \\&lt;br /&gt;
                                &amp;amp;= &lt;br /&gt;
         \left(\begin{array}{cc}&lt;br /&gt;
  \cos(\theta/2) -in_3\sin(\theta/2) &amp;amp; (-in_1-n_2)\sin(\theta/2) \\ &lt;br /&gt;
   (-in_1+n_2)\sin(\theta/2) &amp;amp; \cos(\theta/2) +in_3\sin(\theta/2)  \end{array}\right).&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Let us prove this.  First, using [[Chapter 3 - Physics of Quantum Information#eq3.7|Eq. (3.7)]], we will need to find &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})^m\,\!&amp;lt;/math&amp;gt;, i.e., all powers of &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})\,\!&amp;lt;/math&amp;gt;.  This turns out to be fairly easy.  First note that &amp;lt;math&amp;gt;\hat{n}\cdot \hat{n} = 1\,\!&amp;lt;/math&amp;gt; since &amp;lt;math&amp;gt;\hat{n}\,\!&amp;lt;/math&amp;gt; is a unit vector.  Then note that &lt;br /&gt;
&amp;lt;math&amp;gt;\sigma_i\sigma_j = \mathbb{I}\delta_{ij} +i \epsilon_{ijk}\sigma_k,&amp;lt;/math&amp;gt;.  (See [[Chapter 2 - Qubits and Collections of Qubits#eq2.21|Eq. (2.21)]] as well as Eqs. [[Appendix C - Vectors and Linear Algebra#eqC.17|(C.17)]] and [[Appendix C - Vectors and Linear Algebra#eqC.8|(C.8)]].)  These imply that (recalling that &amp;lt;math&amp;gt; \sigma_x =\sigma_1,\; \sigma_y = \sigma_2, \;\;\sigma_z=\sigma_3 &amp;lt;/math&amp;gt;),&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
(\vec{n}\cdot\vec{\sigma})^2 &amp;amp;= (n_1 \sigma_1 + n_2 \sigma_2 + n_3 \sigma_3)^2 \\&lt;br /&gt;
                  &amp;amp;= \left( \sum_i n_i \sigma_i\right) \left( \sum_j n_j \sigma_j \right) \\&lt;br /&gt;
                  &amp;amp;= \sum_{ij} n_i n_j \sigma_i \sigma_j \\&lt;br /&gt;
                  &amp;amp;= \sum_{ij} n_i n_j (\mathbb{I} \delta_{ij}+i\epsilon_{ijk}\sigma_k ) \\&lt;br /&gt;
                  &amp;amp;= \sum_{ij} n_i n_i   \mathbb{I} = \mathbb{I}.&lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
To get the third line, one just uses [[Chapter 2 - Qubits and Collections of Qubits#eq2.21|Eq. (2.21)]].  To see that &amp;lt;math&amp;gt;\sum_{ij} n_i n_j \epsilon_{ijk}\sigma_k \,\!&amp;lt;/math&amp;gt; is zero, note that &amp;lt;math&amp;gt;n_i n_j  \,\!&amp;lt;/math&amp;gt; is symmetric in &amp;lt;math&amp;gt;i,j\,\!&amp;lt;/math&amp;gt; but &amp;lt;math&amp;gt;\epsilon_{ijk} \,\!&amp;lt;/math&amp;gt; is antisymmetric in &amp;lt;math&amp;gt;i,j\,\!&amp;lt;/math&amp;gt;.  Another way to see this more explicitly is to write out the sum and notice that each term shows up with a + and - sign thus cancelling each other out.  Therefore, all the even powers of &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})\,\!&amp;lt;/math&amp;gt; are just equal to the identity matrix and all odd powers are just &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})\,\!&amp;lt;/math&amp;gt; times the even parts.  Thus the sum in [[Chapter 3 - Physics of Quantum Information#eq3.7|Eq. (3.7)]] reduces to &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\exp(-i \hat{n}\cdot \vec{\sigma}\theta/2) &amp;amp; = \mathbb{I} \cos(\theta/2) + i (\hat{n}\cdot \vec{\sigma})\sin(\theta/2). \;\;\;\; \square&lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Notice that this is a ''special unitary matrix.''  (See Section [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|Unitary Matrices]].)&lt;br /&gt;
To see that this is the most general SU(2) matrix, one needs to verify that any complex &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; unitary matrix can be written in this form.  (One way to do this is to start with a generic matrix and impose the restrictions.  Here one may simply convince oneself that this is general through observation by acting on basis vectors.)  This is the most general qubit transformation and can be interpreted as a rotation about the axis &amp;lt;math&amp;gt;\hat{n}\,\!&amp;lt;/math&amp;gt; by an angle &amp;lt;math&amp;gt;\theta\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
Another parametrization of this set of matrices is the following, called the Euler angle parametrization:&lt;br /&gt;
 {{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.25}}&lt;br /&gt;
In this case the matrices &amp;lt;math&amp;gt;\sigma_z\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\sigma_y\,\!&amp;lt;/math&amp;gt; are not unique.  Any two of the three Pauli matrices (or one of each) may be chosen.  This is quite simple, useful, and generalizable to SU(N) for N arbitrary.  In the simple case of a qubit, one may convince oneself by acting on basis vectors as before.  However, with a little thought, one may see that rotating to a position on the sphere by the first angle, followed by rotations using the other two, will provide for a general orientation of an object.&lt;br /&gt;
&lt;br /&gt;
====Similarity Transformation====&lt;br /&gt;
&lt;br /&gt;
A ''similarity transformation'' &amp;lt;!--\index{similarity transformation}--&amp;gt; &lt;br /&gt;
of an &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; by an invertible matrix &amp;lt;math&amp;gt;S\,\!&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;S A S^{-1}\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
There are (at least) two important things to note about similarity&lt;br /&gt;
transformations: &lt;br /&gt;
#Similarity transformations leave the trace of a matrix unchanged.  This is shown explicitly in [[#The Trace|Section 3.5]].&lt;br /&gt;
#Similarity transformations leave the determinant of a matrix unchanged, or invariant.  This is because &amp;lt;center&amp;gt;&amp;lt;math&amp;gt; \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged.  Let &amp;lt;math&amp;gt;A^\prime = SAS^{-1}\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;B^\prime = SBS^{-1}\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;C^\prime = SCS^{-1}\,\!&amp;lt;/math&amp;gt;.  If &amp;lt;math&amp;gt;AB=C\,\!&amp;lt;/math&amp;gt;, then &amp;lt;math&amp;gt;A^\prime B^\prime = C^\prime\,\!&amp;lt;/math&amp;gt;, since &amp;lt;math&amp;gt;A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} =  SCS^{-1}=C^\prime\,\!&amp;lt;/math&amp;gt;.  The two matrices &amp;lt;math&amp;gt;A^\prime\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; are said to be ''similar''.&lt;br /&gt;
&amp;lt;!-- \index{similar matrices} --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Eigenvalues and Eigenvectors===&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!-- \index{eigenvalues}\index{eigenvectors} --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A matrix can always be diagonalized.  By this, it is meant that for&lt;br /&gt;
every complex matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; there is a diagonal matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; such that &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
M = UDV,  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.26}}&lt;br /&gt;
where &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;V\,\!&amp;lt;/math&amp;gt; are unitary matrices.  This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; are called the ''singular values'' &amp;lt;!--\index{singular values}--&amp;gt; &lt;br /&gt;
of the matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt;.  However, the singular values are not always easy to find.  &lt;br /&gt;
&lt;br /&gt;
For the special case that the matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; is Hermitian &amp;lt;math&amp;gt;(M^\dagger = M)\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
the matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; can be written as&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
M = U D U^\dagger&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.27}}&lt;br /&gt;
where &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; is unitary &amp;lt;math&amp;gt;(U^{-1}=U^\dagger)\,\!&amp;lt;/math&amp;gt;.  In this case the elements&lt;br /&gt;
of the matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; are called ''eigenvalues''. &amp;lt;!--\index{eigenvalues}--&amp;gt;&lt;br /&gt;
Very often eigenvalues are introduced as solutions to the equation&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt;\left\vert v\right\rangle\,\!&amp;lt;/math&amp;gt; is an ''eigenvector''. &amp;lt;!--\index{eigenvector} --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
To find the eigenvalues and eigenvectors of a matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt;, we follow a&lt;br /&gt;
standard procedure which is to calculate&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(\lambda\mathbb{I} - M) = 0&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.28}}&lt;br /&gt;
and then solve for &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  The different solutions for &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt; is the&lt;br /&gt;
set of eigenvalues and is called the ''spectrum''. &amp;lt;!-- \index{spectrum}--&amp;gt; Let the different eigenvalues be denoted by &amp;lt;math&amp;gt;\lambda_i\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;i=1,2,...,n\,\!&amp;lt;/math&amp;gt; fo an &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; vector.  If two&lt;br /&gt;
eigenvalues are equal, we say the spectrum is &lt;br /&gt;
''degenerate''. &amp;lt;!--\index{degenerate}--&amp;gt; To find the&lt;br /&gt;
eigenvectors, which correspond to different eigenvalues, the equation &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
must be solved for each value of &amp;lt;math&amp;gt;i\,\!&amp;lt;/math&amp;gt;.  Notice that this equation&lt;br /&gt;
holds even if we multiply both sides by some complex number.  This&lt;br /&gt;
implies that an eigenvector can always be scaled.  Usually they are&lt;br /&gt;
normalized to obtain an orthonormal set.  As we will see by example,&lt;br /&gt;
degenerate eigenvalues require some care.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Example 1====&lt;br /&gt;
&lt;br /&gt;
Consider a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; Hermitian matrix&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\sigma = \left(\begin{array}{cc} &lt;br /&gt;
               1+a &amp;amp; b-ic \\&lt;br /&gt;
              b+ic &amp;amp; 1-a  \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.29}}&lt;br /&gt;
To find the eigenvalues &amp;lt;!--\index{eigenvalues}--&amp;gt; &lt;br /&gt;
of this, we follow a standard procedure, which&lt;br /&gt;
is to calculate &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(\sigma-\lambda\mathbb{I}) = 0,&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.30}}&lt;br /&gt;
and solve for &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  The eigenvalues of this matrix are given by&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det\left(\begin{array}{cc} &lt;br /&gt;
               1+a-\lambda &amp;amp; b-ic \\&lt;br /&gt;
              b+ic &amp;amp; 1-a-\lambda  \end{array}\right) =0,  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
which implies that the eigenvalues are&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
and the eigenvectors &amp;lt;!--\index{eigenvectors}--&amp;gt; are&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_1=\left(\begin{array}{c}&lt;br /&gt;
        i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ &lt;br /&gt;
        2b &lt;br /&gt;
        \end{array}\right), &lt;br /&gt;
v_2= \left(\begin{array}{c}&lt;br /&gt;
         i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ &lt;br /&gt;
         2b \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
These expressions are useful for calculating properties of qubit&lt;br /&gt;
states as will be seen in the text.&lt;br /&gt;
&lt;br /&gt;
====Example 2====&lt;br /&gt;
&lt;br /&gt;
Now consider a &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; matrix,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
N= \left(\begin{array}{ccc}&lt;br /&gt;
              1 &amp;amp; -i &amp;amp; 0 \\&lt;br /&gt;
              i &amp;amp; 1 &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 0 &amp;amp; 1 \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
First we calculate&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det\left(\begin{array}{ccc}&lt;br /&gt;
              1-\lambda &amp;amp; -i &amp;amp; 0 \\&lt;br /&gt;
              i         &amp;amp; 1-\lambda  &amp;amp; 0 \\&lt;br /&gt;
              0         &amp;amp;       0    &amp;amp; 1-\lambda &lt;br /&gt;
           \end{array}\right) &lt;br /&gt;
    = (1-\lambda)[(1-\lambda)^2-1].&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This implies that the eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']] are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\lambda = 1,0, \mbox{ or } 2.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Let &amp;lt;math&amp;gt;\lambda_1=1\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;\lambda_0 = 0\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;\lambda_2 = 2\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
To find eigenvectors, &amp;lt;!--\index{eigenvalues}--&amp;gt; we calculate&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
Nv &amp;amp;= \lambda v, \\&lt;br /&gt;
\left(\begin{array}{ccc}&lt;br /&gt;
              1 &amp;amp; -i &amp;amp; 0 \\&lt;br /&gt;
              i &amp;amp; 1 &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 0 &amp;amp; 1 \end{array}\right)\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right) &amp;amp;= \lambda\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right)&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.31}}&lt;br /&gt;
for each &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
For &amp;lt;math&amp;gt;\lambda = 1\,\!&amp;lt;/math&amp;gt;, we get the following equations:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_1 -iv_2 &amp;amp;= v_1, \\&lt;br /&gt;
iv_1+v_2 &amp;amp;= v_2,  \\&lt;br /&gt;
v_3 &amp;amp;= v_3. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.32}}&lt;br /&gt;
Solving this obtains &amp;lt;math&amp;gt;v_2 =0\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;v_1 =0\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;v_3\,\!&amp;lt;/math&amp;gt; is any non-zero number (which will be chosen to normalize the vector).  For &amp;lt;math&amp;gt;\lambda&lt;br /&gt;
=0\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_1  &amp;amp;= iv_2, \\&lt;br /&gt;
v_3 &amp;amp;= 0. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.33}}&lt;br /&gt;
And finally, for &amp;lt;math&amp;gt;\lambda = 2\,\!&amp;lt;/math&amp;gt;, we obtain&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_1 -iv_2 &amp;amp;= 2v_1, \\&lt;br /&gt;
iv_1+v_2 &amp;amp;= 2v_2, \\&lt;br /&gt;
v_3 &amp;amp;= 2v_3, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.34}}&lt;br /&gt;
so that &amp;lt;math&amp;gt;v_1 = -iv_2\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
Therefore, our three eigenvectors &amp;lt;!--\index{eigenvalues}--&amp;gt; are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; &lt;br /&gt;
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; &lt;br /&gt;
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The matrix &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}&lt;br /&gt;
              i/\sqrt{2} &amp;amp; 0     &amp;amp; -i/\sqrt{2} \\&lt;br /&gt;
              1/\sqrt{2} &amp;amp; 0     &amp;amp; 1/\sqrt{2} \\&lt;br /&gt;
              0          &amp;amp; 1     &amp;amp; 0 \end{array}\right)&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
is the matrix that diagonalizes &amp;lt;math&amp;gt;N\,\!&amp;lt;/math&amp;gt; in the following way:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
N = VDV^\dagger&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
D = \left(\begin{array}{ccc}&lt;br /&gt;
              0 &amp;amp; 0  &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 1  &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 0  &amp;amp; 2 \end{array}\right)&lt;br /&gt;
.\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
We may write this as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
V^\dagger N V = D.   &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This is sometimes called the ''eigenvalue decompostion''&amp;lt;!--\index{eigenvalue decomposition}--&amp;gt;  of the matrix and can also be written as&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
N = \sum_i \lambda_i v_iv^\dagger_i.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.35}}&lt;br /&gt;
&lt;br /&gt;
====Example 3====&lt;br /&gt;
&lt;br /&gt;
Next, consider the complex &amp;lt;math&amp;gt;3\times 3&amp;lt;/math&amp;gt; Hermitian matrix &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
M = \left(\begin{array}{ccc}&lt;br /&gt;
              \frac{5}{2} &amp;amp; 0  &amp;amp; \frac{i}{2} \\&lt;br /&gt;
              0 &amp;amp; 2  &amp;amp; 0 \\&lt;br /&gt;
              -\frac{i}{2} &amp;amp; 0  &amp;amp; \frac{5}{2} \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
First we calculate&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det\left(\begin{array}{ccc}&lt;br /&gt;
              \frac{5}{2}-\lambda &amp;amp; 0 &amp;amp; \frac{i}{2} \\&lt;br /&gt;
              0         &amp;amp; 2-\lambda  &amp;amp; 0 \\&lt;br /&gt;
              -\frac{i}{2}         &amp;amp;       0    &amp;amp; \frac{5}{2}-\lambda &lt;br /&gt;
           \end{array}\right) &lt;br /&gt;
    = (2-\lambda)\left[\left(\frac{5}{2}-\lambda\right)^2-\frac{1}{4}\right].&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This implies that the eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']] are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\lambda = 2,2, \mbox{ or } 3.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Note that there are two that are the same, or degenerate.  &lt;br /&gt;
Let &amp;lt;math&amp;gt;\lambda_1=2\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;\lambda_2 = 2\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;\lambda_3 = 3\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
To find eigenvectors, &amp;lt;!--\index{eigenvalues}--&amp;gt; we calculate&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
Mv &amp;amp;= \lambda v, \\&lt;br /&gt;
\left(\begin{array}{ccc}&lt;br /&gt;
              \frac{5}{2} &amp;amp; 0 &amp;amp; \frac{i}{2} \\&lt;br /&gt;
              0 &amp;amp; 2 &amp;amp; 0 \\&lt;br /&gt;
              -\frac{i}{2} &amp;amp; 0 &amp;amp; \frac{5}{2} \end{array}\right)\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right) &amp;amp;= \lambda\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right)&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.36}}&lt;br /&gt;
for each &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
For &amp;lt;math&amp;gt;\lambda = 3\,\!&amp;lt;/math&amp;gt;, we get the following equations:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\frac{5}{2}v_1 + \frac{i}{2}v_3 &amp;amp;= 3v_1, \\&lt;br /&gt;
2v_2 &amp;amp;= 3v_2,  \\&lt;br /&gt;
-\frac{i}{2}v_1 + \frac{5}{2}v_3 &amp;amp;= 3v_3, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.37}}&lt;br /&gt;
so &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
iv_3  &amp;amp;= v_1, \\&lt;br /&gt;
v_2 &amp;amp;= 0. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.38}}&lt;br /&gt;
Now for &amp;lt;math&amp;gt;\lambda = 2\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\frac{5}{2}v_1 +\frac{i}{2}v_3 &amp;amp;= 2 v_1, \\&lt;br /&gt;
2v_2 &amp;amp;= 2v_2, \\&lt;br /&gt;
-\frac{i}{2}v_1 + \frac{5}{2}v_3 &amp;amp;= 2v_3, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.39}}&lt;br /&gt;
so &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_3  &amp;amp;= iv_1, \\&lt;br /&gt;
v_2 &amp;amp;= \mbox{anything}. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.40}}&lt;br /&gt;
We would like to have a set of orthonormal vectors.  (We can always choose the set to be orthonormal.)  We choose the three eigenvectors to be&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_2 = \left(\begin{array}{c} 1 \\ a \\ i&lt;br /&gt;
  \end{array}\right), \;\;  &lt;br /&gt;
v_2^\prime = \left(\begin{array}{c} 1 \\ a^\prime \\ i&lt;br /&gt;
  \end{array}\right), \;\;&lt;br /&gt;
v_3 = \left(\begin{array}{c} i \\ 0 \\ 1&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
We set the inner product of the two vectors &amp;lt;math&amp;gt; v_2\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; v_2^\prime \,\!&amp;lt;/math&amp;gt; equal to zero so as to have then be orthogonal: &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
1 + a a^\prime +1 = 2 + a a^\prime = 0.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Now we can choose &amp;lt;math&amp;gt; a = \sqrt{2}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; a^\prime = -\sqrt{2}\,\!&amp;lt;/math&amp;gt; so that the normalized eigenvectors are&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_2 = \frac{1}{2}\left(\begin{array}{c} 1 \\ \sqrt{2} \\ i&lt;br /&gt;
  \end{array}\right), \;\;  &lt;br /&gt;
v_2^\prime = \frac{1}{2}\left(\begin{array}{c} 1 \\ -\sqrt{2} \\ i&lt;br /&gt;
  \end{array}\right), \;\;&lt;br /&gt;
v_3 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 0 \\ 1&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Tensor Products===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The tensor product, &amp;lt;!--\index{tensor product} --&amp;gt;&lt;br /&gt;
or the Kronecker product, &amp;lt;!--\index{Kronecker product}--&amp;gt;&lt;br /&gt;
is used extensively in quantum mechanics and&lt;br /&gt;
throughout the course.  It is commonly denoted with a &amp;lt;math&amp;gt;\otimes\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
symbol, although this is often left out.  In fact, the following&lt;br /&gt;
are commonly found in the literature as notation for the tensor&lt;br /&gt;
product of two vectors &amp;lt;math&amp;gt;\left\vert\Psi\right\rangle\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left\vert\Phi\right\rangle\,\!&amp;lt;/math&amp;gt;:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &amp;amp;= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle  \\&lt;br /&gt;
                         &amp;amp;= \left\vert\Psi\Phi\right\rangle.&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.41}}&lt;br /&gt;
Each of these has its advantages and will all be used in&lt;br /&gt;
different circumstances in this text.  &lt;br /&gt;
&lt;br /&gt;
The tensor product is also often used for operators.  Several&lt;br /&gt;
examples &lt;br /&gt;
will be given, one that explicitly calculates the tensor product for&lt;br /&gt;
two vectors and one that calculates it for two matrices which could&lt;br /&gt;
represent operators.  However, these are not different in the sense&lt;br /&gt;
that a vector is a &amp;lt;math&amp;gt;1\times n\,\!&amp;lt;/math&amp;gt; or an &amp;lt;math&amp;gt;n\times 1\,\!&amp;lt;/math&amp;gt; matrix.  It is also&lt;br /&gt;
noteworthy that the two objects in the tensor product need not be of&lt;br /&gt;
the same type.  In general, a tensor product of an &amp;lt;math&amp;gt;n\times m\,\!&amp;lt;/math&amp;gt; object&lt;br /&gt;
(array) with a &amp;lt;math&amp;gt;p\times q\,\!&amp;lt;/math&amp;gt; object will produce an &amp;lt;math&amp;gt;np\times mq\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
object.  &lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
The tensor product of two objects is computed as follows.&lt;br /&gt;
Let &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; be an &amp;lt;math&amp;gt;n\times m\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt; be a &amp;lt;math&amp;gt;p\times q\,\!&amp;lt;/math&amp;gt; array, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
A = \left(\begin{array}{cccc} &lt;br /&gt;
           a_{11} &amp;amp; a_{12} &amp;amp; \cdots &amp;amp; a_{1m} \\&lt;br /&gt;
           a_{21} &amp;amp; a_{22} &amp;amp; \cdots &amp;amp; a_{2m} \\&lt;br /&gt;
           \vdots &amp;amp;        &amp;amp; \ddots &amp;amp;      \\&lt;br /&gt;
           a_{n1} &amp;amp; a_{n2} &amp;amp; \cdots &amp;amp; a_{nm} \end{array}\right),&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.42}}&lt;br /&gt;
and similarly for &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt;.  Then &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
A\otimes B = \left(\begin{array}{cccc} &lt;br /&gt;
             a_{11}B &amp;amp; a_{12}B &amp;amp; \cdots &amp;amp; a_{1m}B \\&lt;br /&gt;
             a_{21}B &amp;amp; a_{22}B &amp;amp; \cdots &amp;amp; a_{2m}B \\&lt;br /&gt;
             \vdots  &amp;amp;         &amp;amp; \ddots &amp;amp;      \\&lt;br /&gt;
             a_{n1}B &amp;amp; a_{n2}B &amp;amp; \cdots &amp;amp; a_{nm}B \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.43}}&lt;br /&gt;
&lt;br /&gt;
Let us now consider two examples.  First let &amp;lt;math&amp;gt;\left\vert\phi\right\rangle\,\!&amp;lt;/math&amp;gt; and&lt;br /&gt;
&amp;lt;math&amp;gt;\left\vert\psi\right\rangle\,\!&amp;lt;/math&amp;gt; be as before,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right) \;\; &lt;br /&gt;
\mbox{and} \;\; &lt;br /&gt;
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta&lt;br /&gt;
  \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Then &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &amp;amp;= \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right) &lt;br /&gt;
\otimes &lt;br /&gt;
\left(\begin{array}{c} \gamma \\ \delta&lt;br /&gt;
  \end{array}\right)   \\&lt;br /&gt;
                            &amp;amp;= \left(\begin{array}{c} \alpha\gamma\\ &lt;br /&gt;
                                                     \alpha\delta \\&lt;br /&gt;
                                                     \beta\gamma \\ &lt;br /&gt;
                                                     \beta\delta &lt;br /&gt;
                                       \end{array}\right).  &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.44}}&lt;br /&gt;
Also&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &amp;amp;= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\&lt;br /&gt;
                            &amp;amp;= \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right) &lt;br /&gt;
\otimes &lt;br /&gt;
\left(\begin{array}{cc} \gamma^* &amp;amp; \delta^*&lt;br /&gt;
  \end{array}\right)   \\&lt;br /&gt;
                            &amp;amp;= \left(\begin{array}{cc}&lt;br /&gt;
                                \alpha\gamma^* &amp;amp; \alpha\delta^* \\&lt;br /&gt;
                                \beta\gamma^*  &amp;amp; \beta\delta^* &lt;br /&gt;
                                       \end{array}\right).  &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.45}}&lt;br /&gt;
&lt;br /&gt;
Now consider two matrices&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
A = \left(\begin{array}{cc} &lt;br /&gt;
                 a &amp;amp; b \\&lt;br /&gt;
                 c &amp;amp; d  \end{array}\right) &lt;br /&gt;
 \;\; \mbox{and} \;\;&lt;br /&gt;
B = \left(\begin{array}{cc} &lt;br /&gt;
               e &amp;amp; f \\&lt;br /&gt;
               g &amp;amp; h  \end{array}\right).  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Then &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
A\otimes B &amp;amp;=  \left(\begin{array}{cc} &lt;br /&gt;
                 a &amp;amp; b \\&lt;br /&gt;
                 c &amp;amp; d  \end{array}\right) &lt;br /&gt;
 \otimes&lt;br /&gt;
               \left(\begin{array}{cc} &lt;br /&gt;
                 e &amp;amp; f \\&lt;br /&gt;
                 g &amp;amp; h  \end{array}\right)   \\  &lt;br /&gt;
           &amp;amp;=  \left(\begin{array}{cccc} &lt;br /&gt;
                 ae &amp;amp; af &amp;amp; be &amp;amp; bf \\&lt;br /&gt;
                 ag &amp;amp; ah &amp;amp; bg &amp;amp; bh \\&lt;br /&gt;
                 ce &amp;amp; cf &amp;amp; de &amp;amp; df \\&lt;br /&gt;
                 cg &amp;amp; ch &amp;amp; dg &amp;amp; dh \end{array}\right).  &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.46}}&lt;br /&gt;
&lt;br /&gt;
====Properties of Tensor Products====&lt;br /&gt;
&lt;br /&gt;
Listed here are properties of tensor products that are useful, with &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;C\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; of any type:&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)(C\otimes D) = AC \otimes BD\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)^T = A^T\otimes B^T\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)^* = A^*\otimes B^*\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A+B) \otimes C = A\otimes C+B\otimes C\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;A\otimes(B+C) = A\otimes B + A\otimes C\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
(See [[Bibliography#HornNJohnsonII|Horn and Johnson, Topics in Matrix Analysis [10]]], Chapter 4.)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
	<entry>
		<id>https://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&amp;diff=2354</id>
		<title>Appendix C - Vectors and Linear Algebra</title>
		<link rel="alternate" type="text/html" href="https://www2.physics.siu.edu/qunet/wiki/index.php?title=Appendix_C_-_Vectors_and_Linear_Algebra&amp;diff=2354"/>
		<updated>2013-09-17T01:20:47Z</updated>

		<summary type="html">&lt;p&gt;Hilary: /* Linear Algebra: Matrices */&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;===Introduction===&lt;br /&gt;
&lt;br /&gt;
This appendix introduces some aspects of linear algebra and complex&lt;br /&gt;
algebra that will be helpful for the course.  In addition, Dirac&lt;br /&gt;
notation is introduced and explained.&lt;br /&gt;
&lt;br /&gt;
===Vectors===&lt;br /&gt;
&lt;br /&gt;
Here we review some facts about real vectors before discussing the representation and complex analogues used in quantum mechanics.  &lt;br /&gt;
&lt;br /&gt;
====Real Vectors====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The simple definition of a vector --- an object that has magnitude and&lt;br /&gt;
direction --- is helpful to keep in mind even when dealing with complex&lt;br /&gt;
and/or abstract vectors as we will here.  In three dimensional space,&lt;br /&gt;
a vector is often written as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z},&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where the hat (&amp;lt;math&amp;gt;\hat{\cdot}\,\!&amp;lt;/math&amp;gt;) denotes a unit vector and the components&lt;br /&gt;
&amp;lt;math&amp;gt;v_i\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;i = x,y,z\,\!&amp;lt;/math&amp;gt; are just numbers.  The unit vectors are also&lt;br /&gt;
known as ''basis'' vectors. &amp;lt;!-- \index{basis vectors!real} --&amp;gt; &lt;br /&gt;
This is because any vector&lt;br /&gt;
in real three-dimensional space can be written in terms of these unit/basis vectors.  In&lt;br /&gt;
some sense they are the basic components of any vector.  Other basis&lt;br /&gt;
vectors could be used, however, such as in spherical and cylindrical coordinates.  When dealing with more abstract and/or complex vectors,&lt;br /&gt;
it is often helpful to ask what one would do for an ordinary&lt;br /&gt;
three-dimensional vector.  For example, properties of unit vectors,&lt;br /&gt;
dot products, etc. in three-dimensions are similar to the analogous&lt;br /&gt;
constructions in higher dimensions.  &lt;br /&gt;
&lt;br /&gt;
The ''inner product'',&amp;lt;!-- \index{inner product}--&amp;gt; or ''dot product'',&amp;lt;!--\index{dot product}--&amp;gt; for two real three-dimensional vectors,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v} = v_x\hat{x} + v_y \hat{y} + v_z\hat{z}, \;\; &lt;br /&gt;
\vec{w} = w_x\hat{x} + w_y \hat{y} + w_z\hat{z},&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
can be computed as follows:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v}\cdot\vec{w} = v_xw_x + v_yw_y + v_zw_z.&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
For the inner product of &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt; with itself, we get the square of&lt;br /&gt;
the magnitude of &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt;, denoted &amp;lt;math&amp;gt;|\vec{v}|^2\,\!&amp;lt;/math&amp;gt;:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
|\vec{v}|^2 = \vec{v}\cdot\vec{v} = v_xv_x + v_yv_y +&lt;br /&gt;
v_zv_z=v_x^2+v_y^2+v_z^2. &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If we want a unit vector in the direction of &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt;, we can simply divide it&lt;br /&gt;
by its magnitude:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\hat{v} = \frac{\vec{v}}{|\vec{v}|}.  &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Now, of course, &amp;lt;math&amp;gt;\hat{v}\cdot\hat{v}= 1\,\!&amp;lt;/math&amp;gt;, which can easily be checked.  &lt;br /&gt;
&lt;br /&gt;
There are several ways to represent a vector.  The ones we will use&lt;br /&gt;
most often are column and row vector notations.  So, for example, we&lt;br /&gt;
could write the vector above as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\vec{v} = \left(\begin{array}{c} v_x \\ v_y \\ v_z&lt;br /&gt;
  \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
In this case, our unit vectors are represented by the following: &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\hat{x} = \left(\begin{array}{c} 1 \\ 0 \\ 0&lt;br /&gt;
  \end{array}\right), \;\;  &lt;br /&gt;
\hat{y} = \left(\begin{array}{c} 0 \\ 1 \\ 0&lt;br /&gt;
  \end{array}\right), \;\;&lt;br /&gt;
\hat{z} = \left(\begin{array}{c} 0 \\ 0 \\ 1&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
We next turn to the subject of complex vectors and the relevant&lt;br /&gt;
notation. &lt;br /&gt;
We will see how to compute the inner product later, since some other&lt;br /&gt;
definitions are required.&lt;br /&gt;
&lt;br /&gt;
====Complex Vectors====&lt;br /&gt;
&lt;br /&gt;
For complex vectors in quantum mechanics, Dirac notation is used most often.  This notation uses a &amp;lt;math&amp;gt;\left\vert \cdot \right\rangle\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
called a ''ket'', for a vector.  So our vector &amp;lt;math&amp;gt;\vec{v}\,\!&amp;lt;/math&amp;gt; would be&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert v \right\rangle  = \left(\begin{array}{c} v_x \\ v_y \\ v_z&lt;br /&gt;
  \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
For qubits, i.e. two-state quantum systems, complex vectors will often be used:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align} \left\vert \psi \right\rangle &amp;amp;= \left(\begin{array}{c} \alpha \\ \beta &lt;br /&gt;
  \end{array}\right) \\&lt;br /&gt;
           &amp;amp;=\alpha \left\vert 0\right\rangle + \beta\left\vert 1\right\rangle,\end{align}&amp;lt;/math&amp;gt;|C.1}} &lt;br /&gt;
where&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert 0\right\rangle = \left(\begin{array}{c} 1 \\ 0 &lt;br /&gt;
  \end{array}\right), \;\;\mbox{and} \;\;&lt;br /&gt;
\left\vert 1\right\rangle = \left(\begin{array}{c} 0 \\ 1 &lt;br /&gt;
  \end{array}\right)&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
are the basis vectors.  The two numbers &amp;lt;math&amp;gt;\alpha\,\!&amp;lt;/math&amp;gt; and&lt;br /&gt;
&amp;lt;math&amp;gt;\beta\,\!&amp;lt;/math&amp;gt; are complex numbers, so the vector is said to&lt;br /&gt;
be a complex vector.&lt;br /&gt;
&lt;br /&gt;
====Inner Product====&lt;br /&gt;
&lt;br /&gt;
Now let us suppose we have another complex vector,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert \phi \right\rangle  = \left(\begin{array}{c} \gamma \\ \delta &lt;br /&gt;
  \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The ''inner product'' between two vectors is often written as &amp;lt;math&amp;gt;\left\langle \phi \vert \psi \right\rangle \;\! &amp;lt;/math&amp;gt;, which is the same as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\begin{align} (\left\vert \phi \right\rangle )^\dagger\left\vert \psi \right\rangle &lt;br /&gt;
&amp;amp;= \left(\begin{array}{c} \gamma \\ \delta \end{array}\right)^\dagger&lt;br /&gt;
\left(\begin{array}{c} \alpha \\ \beta   \end{array}\right) \\&lt;br /&gt;
           &amp;amp;= \left(\begin{array}{cc} \gamma^* &amp;amp; \delta^* \end{array}\right) \left(\begin{array}{c} \alpha \\ \beta   \end{array}\right) \\  &lt;br /&gt;
           &amp;amp;= \gamma^*\alpha + \delta^*\beta \end{align} \;\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Outer Product====&lt;br /&gt;
&lt;br /&gt;
The ''outer product'' between these same two vectors is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt; &lt;br /&gt;
\begin{align} (\left\vert \phi \right\rangle )(\left\vert \psi \right\rangle)^\dagger &lt;br /&gt;
 &amp;amp;=  \left\vert \phi \right\rangle \left\langle \psi \right\vert \\&lt;br /&gt;
&amp;amp;= \left(\begin{array}{c} \gamma \\ \delta \end{array}\right)&lt;br /&gt;
\left(\begin{array}{c} \alpha \\ \beta   \end{array}\right)^\dagger \\&lt;br /&gt;
           &amp;amp;= \left(\begin{array}{c} \gamma \\ \delta \end{array}\right) \left(\begin{array}{cc} \alpha^* &amp;amp; \beta^*   \end{array}\right) \\  &lt;br /&gt;
           &amp;amp;=   \left(\begin{array}{cc} \gamma\alpha^* &amp;amp; \gamma\beta^* \\  \delta\alpha^* &amp;amp; \delta\beta^*  \end{array}\right) \end{align}\;\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Linear Algebra: Matrices===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are many aspects of linear algebra that are quite useful in&lt;br /&gt;
quantum mechanics.  We will briefly discuss several of these aspects here.&lt;br /&gt;
First, some definitions and properties are provided that will&lt;br /&gt;
be useful.  Some familiarity with matrices&lt;br /&gt;
will be assumed, although many basic definitions are also included.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Let us denote some &amp;lt;math&amp;gt;m\times n\,\!&amp;lt;/math&amp;gt; matrix by &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt;.  The set of all &amp;lt;math&amp;gt;m\times&lt;br /&gt;
n\,\!&amp;lt;/math&amp;gt; matrices with real entries is &amp;lt;math&amp;gt;M(n\times m,\mathbb{R})\,\!&amp;lt;/math&amp;gt;.  Such matrices&lt;br /&gt;
are said to be real since they have all real entries.  Similarly, the&lt;br /&gt;
set of &amp;lt;math&amp;gt;m\times n\,\!&amp;lt;/math&amp;gt; complex matrices is &amp;lt;math&amp;gt;M(m\times n,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.  For the&lt;br /&gt;
set of square &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; complex matrices, we simply write&lt;br /&gt;
&amp;lt;math&amp;gt;M(n,\mathbb{C})\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
We will also refer to the set of matrix elements, &amp;lt;math&amp;gt;a_{ij}\,\!&amp;lt;/math&amp;gt;, where the&lt;br /&gt;
first index (&amp;lt;math&amp;gt;i\,\!&amp;lt;/math&amp;gt; in this case) labels the row and the second &amp;lt;math&amp;gt;(j)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
labels the column.  Thus the element &amp;lt;math&amp;gt;a_{23}\,\!&amp;lt;/math&amp;gt; is the element in the&lt;br /&gt;
second row and third column.  A comma is inserted if there is some&lt;br /&gt;
ambiguity.  For example, in a large matrix the element in the&lt;br /&gt;
2nd row and 12th&lt;br /&gt;
column is written as &amp;lt;math&amp;gt;a_{2,12}\,\!&amp;lt;/math&amp;gt; to distinguish between the&lt;br /&gt;
21st row and 2nd column.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Complex Conjugate====&lt;br /&gt;
&lt;br /&gt;
The ''complex conjugate of a matrix'' &amp;lt;!-- \index{complex conjugate!of a matrix}--&amp;gt;&lt;br /&gt;
is the matrix with each element replaced by its complex conjugate.  In&lt;br /&gt;
other words, to take the complex conjugate of a matrix, one takes the&lt;br /&gt;
complex conjugate of each entry in the matrix.  We denote the complex&lt;br /&gt;
conjugate with a star, like this: &amp;lt;math&amp;gt;A^*\,\!&amp;lt;/math&amp;gt;.  For example,&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
A^* &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11} &amp;amp; a_{12} &amp;amp; a_{13} \\&lt;br /&gt;
        a_{21} &amp;amp; a_{22} &amp;amp; a_{23} \\&lt;br /&gt;
        a_{31} &amp;amp; a_{32} &amp;amp; a_{33} \end{array}\right)^*  \\&lt;br /&gt;
    &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11}^* &amp;amp; a_{12}^* &amp;amp; a_{13}^* \\&lt;br /&gt;
        a_{21}^* &amp;amp; a_{22}^* &amp;amp; a_{23}^* \\&lt;br /&gt;
        a_{31}^* &amp;amp; a_{32}^* &amp;amp; a_{33}^* \end{array}\right). &lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.2}}&lt;br /&gt;
(Notice that the notation for a matrix is a capital letter, whereas&lt;br /&gt;
the entries are represented by lower case&lt;br /&gt;
letters.)&lt;br /&gt;
&lt;br /&gt;
====Transpose====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The ''transpose'' &amp;lt;!-- \index{transpose} --&amp;gt; of a matrix is the same set of&lt;br /&gt;
elements, but now the first row becomes the first column, the second row&lt;br /&gt;
becomes the second column, and so on.  Thus the rows and columns are&lt;br /&gt;
interchanged.  For example, for a square &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; matrix, the&lt;br /&gt;
transpose is given by&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
A^T &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11} &amp;amp; a_{12} &amp;amp; a_{13} \\&lt;br /&gt;
        a_{21} &amp;amp; a_{22} &amp;amp; a_{23} \\&lt;br /&gt;
        a_{31} &amp;amp; a_{32} &amp;amp; a_{33} \end{array}\right)^T \\&lt;br /&gt;
    &amp;amp;=&amp;amp; \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11} &amp;amp; a_{21} &amp;amp; a_{31} \\&lt;br /&gt;
        a_{12} &amp;amp; a_{22} &amp;amp; a_{32} \\&lt;br /&gt;
        a_{13} &amp;amp; a_{23} &amp;amp; a_{33} \end{array}\right). &lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.3}}&lt;br /&gt;
&lt;br /&gt;
====Hermitian Conjugate====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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,&lt;br /&gt;
(&amp;lt;math&amp;gt;\dagger\,\!&amp;lt;/math&amp;gt;):&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
(A^T)^* = (A^*)^T \equiv A^\dagger.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.4}}&lt;br /&gt;
For our &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; example, &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
A^\dagger = \left(\begin{array}{ccc}&lt;br /&gt;
        a_{11}^* &amp;amp; a_{21}^* &amp;amp; a_{31}^* \\&lt;br /&gt;
        a_{12}^* &amp;amp; a_{22}^* &amp;amp; a_{32}^* \\&lt;br /&gt;
        a_{13}^* &amp;amp; a_{23}^* &amp;amp; a_{33}^* \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If a matrix is its own Hermitian conjugate, i.e. &amp;lt;math&amp;gt;A^\dagger = A\,\!&amp;lt;/math&amp;gt;, then&lt;br /&gt;
we call it a ''Hermitian matrix''.  &amp;lt;!-- \index{Hermitian matrix}--&amp;gt;&lt;br /&gt;
(Clearly this is only possible for square matrices.) Hermitian&lt;br /&gt;
matrices are very important in quantum mechanics since their&lt;br /&gt;
eigenvalues are real.  (See Sec.([[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|Eigenvalues and Eigenvectors]]).)&lt;br /&gt;
&lt;br /&gt;
====Index Notation====&lt;br /&gt;
&lt;br /&gt;
Very often we write the product of two matrices &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt; simply as&lt;br /&gt;
&amp;lt;math&amp;gt;AB\,\!&amp;lt;/math&amp;gt; and let &amp;lt;math&amp;gt;C=AB\,\!&amp;lt;/math&amp;gt;.  However, it is also quite useful to write this&lt;br /&gt;
in component form.  In this case, if these are &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrices, the component form will be &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
c_{ik} = \sum_{j=1}^n a_{ij}b_{jk}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This says that the element in the &amp;lt;math&amp;gt;i^{\mbox{th}}\,\!&amp;lt;/math&amp;gt; row and&lt;br /&gt;
&amp;lt;math&amp;gt;j^{\mbox{th}}\,\!&amp;lt;/math&amp;gt; column of the matrix &amp;lt;math&amp;gt;C\,\!&amp;lt;/math&amp;gt; is the sum &amp;lt;math&amp;gt;\sum_{j=1}^n&lt;br /&gt;
a_{ij}b_{jk}\,\!&amp;lt;/math&amp;gt;.  The transpose of &amp;lt;math&amp;gt;C\,\!&amp;lt;/math&amp;gt; has elements&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
c_{ki} = \sum_{j=1}^n a_{kj}b_{ji}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Now if we were to transpose &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt; as well, this would read&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
c_{ki} = \sum_{j=1}^n (a_{jk})^T (b_{ij})^T = \sum_{j=1}^n b^T_{ij} a^T_{jk}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This gives us a way of seeing the general rule that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
C^T = B^TA^T.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
It follows that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
C^\dagger = B^\dagger A^\dagger.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====The Trace====&lt;br /&gt;
&lt;br /&gt;
The ''trace'' &amp;lt;!-- \index{trace}--&amp;gt; of a matrix is the sum of the diagonal&lt;br /&gt;
elements and is denoted &amp;lt;math&amp;gt;\mbox{Tr}\,\!&amp;lt;/math&amp;gt;.  So for example, the trace of an&lt;br /&gt;
&amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mbox{Tr}(A) = \sum_{i=1}^n a_{ii}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;.&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Some useful properties of the trace are the following:&lt;br /&gt;
#&amp;lt;math&amp;gt;\mbox{Tr}(AB) = \mbox{Tr}(BA)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;\mbox{Tr}(A + B) = \mbox{Tr}(A) + \mbox{Tr}(B)\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
Using the first of these results,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mbox{Tr}(UAU^{-1}) = \mbox{Tr}(U^{-1}UA) = \mbox{Tr}(A).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This relation is used so often that we state it here explicitly.&lt;br /&gt;
&lt;br /&gt;
====The Determinant====&lt;br /&gt;
&lt;br /&gt;
For a square matrix, the determinant is quite a useful thing.  For&lt;br /&gt;
example, an &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrix is invertible if and only if its&lt;br /&gt;
determinant is not zero.  So let us define the determinant and give&lt;br /&gt;
some properties and examples.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The ''determinant''&amp;lt;!--\index{determinant}--&amp;gt; of a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; matrix, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
N = \left(\begin{array}{cc}&lt;br /&gt;
                 a &amp;amp; b \\&lt;br /&gt;
                 c &amp;amp; d \end{array}\right),&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.5}}&lt;br /&gt;
is given by&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(N) = ad-bc.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.6}}&lt;br /&gt;
Higher-order determinants can be written in terms of smaller ones in a recursive way.  For example, let &lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
M = \left(\begin{array}{ccc}&lt;br /&gt;
                 m_{11} &amp;amp; m_{12} &amp;amp; m_{13} \\&lt;br /&gt;
                 m_{21} &amp;amp; m_{22} &amp;amp; m_{23} \\&lt;br /&gt;
                 m_{31} &amp;amp; m_{32} &amp;amp; m_{33}\end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;  &lt;br /&gt;
Then &lt;br /&gt;
&amp;lt;center&amp;gt;&lt;br /&gt;
&amp;lt;math&amp;gt;&lt;br /&gt;
\det(M) = m_{11}\det\left(\begin{array}{cc}&lt;br /&gt;
                 m_{21} &amp;amp; m_{22} \\&lt;br /&gt;
                 m_{31} &amp;amp; m_{32} \end{array}\right) &lt;br /&gt;
- m_{12}\det\left(\begin{array}{cc}&lt;br /&gt;
                 m_{21} &amp;amp; m_{23} \\&lt;br /&gt;
                 m_{31} &amp;amp; m_{33} \end{array}\right)&lt;br /&gt;
+m_{13}\det\left(\begin{array}{cc}&lt;br /&gt;
                 m_{22} &amp;amp; m_{23} \\&lt;br /&gt;
                 m_{32} &amp;amp; m_{33} \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
&amp;lt;/center&amp;gt;  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The ''determinant''&amp;lt;!-- \index{determinant}--&amp;gt; of a matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; can be&lt;br /&gt;
also be written in terms of its components as &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(A) = \sum_{i,j,k,l,...} \epsilon_{ijkl...}&lt;br /&gt;
a_{1i}a_{2j}a_{3k}a_{4l} ...,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.7}}&lt;br /&gt;
where the symbol &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijkl...} = \begin{cases}&lt;br /&gt;
                       +1, \; \mbox{if } \; ijkl... = 1234... (\mbox{in order, or any even number of permutations}),\\&lt;br /&gt;
                       -1, \; \mbox{if } \; ijkl... = 2134... (\mbox{or any odd number of permutations}),\\&lt;br /&gt;
                       \;\;\; 0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}).&lt;br /&gt;
                      \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.8}}&lt;br /&gt;
&lt;br /&gt;
Let us consider the example of the &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; given&lt;br /&gt;
above.  The determinant can be calculated by&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(A) = \sum_{i,j,k} \epsilon_{ijk}&lt;br /&gt;
a_{1i}a_{2j}a_{3k},&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where, explicitly, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\epsilon_{ijk} = \begin{cases}&lt;br /&gt;
                       +1, \;\mbox{if }\; ijk= 123,231,\; \mbox{or}\; 312, (\mbox{These are even permutations of }123),\\&lt;br /&gt;
                       -1, \;\mbox{if }\; ijk = 213,132,\;\mbox{or}\;321(\mbox{These are odd permuations of }123),\\&lt;br /&gt;
                    \;\;\;  0, \; \mbox{otherwise}, \; (\mbox{meaning any index is repeated}),&lt;br /&gt;
                 \end{cases}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.9}}&lt;br /&gt;
so that &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\det(A) &amp;amp;=&amp;amp; \epsilon_{123}a_{11}a_{22}a_{33} &lt;br /&gt;
         +\epsilon_{132}a_{11}a_{23}a_{32}&lt;br /&gt;
         +\epsilon_{231}a_{12}a_{23}a_{31}  \\&lt;br /&gt;
       &amp;amp;&amp;amp;+\epsilon_{213}a_{12}a_{21}a_{33}&lt;br /&gt;
         +\epsilon_{312}a_{13}a_{21}a_{32}&lt;br /&gt;
         +\epsilon_{321}a_{13}a_{22}a_{31}.&lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.10}}&lt;br /&gt;
Now given the values of &amp;lt;math&amp;gt;\epsilon_{ijk}\,\!&amp;lt;/math&amp;gt; in [[#eqC.9|Eq. C.9]],&lt;br /&gt;
this is&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(A) = a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32} + a_{12}a_{23}a_{31} &lt;br /&gt;
         - a_{12}a_{21}a_{33} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The determinant has several properties that are useful to know.  A few are listed here:  &lt;br /&gt;
#The determinant of the transpose of a matrix is the same as the determinant of the matrix itself: &amp;lt;center&amp;gt;&amp;lt;math&amp;gt; \det(A) = \det(A^T).\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
#The determinant of a product is the product of determinants:    &amp;lt;center&amp;gt;&amp;lt;math&amp;gt; \det(AB) = \det(A)\det(B).\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
From this last property, another specific property can be derived.&lt;br /&gt;
If we take the determinant of the product of a matrix and its&lt;br /&gt;
inverse, we find&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(U U^{-1}) = \det(U)\det(U^{-1}) = \det(\mathbb{I}) = 1,&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
since the determinant of the identity is one.  This implies that&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det(U^{-1}) = \frac{1}{\det(U)}.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====The Inverse of a Matrix====&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The inverse &amp;lt;!-- \index{inverse}--&amp;gt; of a square matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; is another matrix,&lt;br /&gt;
denoted &amp;lt;math&amp;gt;A^{-1}\,\!&amp;lt;/math&amp;gt;, such that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
AA^{-1} = A^{-1}A = \mathbb{I},&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt;\mathbb{I}\,\!&amp;lt;/math&amp;gt; is the identity matrix consisting of zeroes everywhere&lt;br /&gt;
except the diagonal, which has ones.  For example, the &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
identity matrix is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mathbb{I}_3 = \left(\begin{array}{ccc} 1 &amp;amp; 0 &amp;amp; 0 \\ 0 &amp;amp; 1 &amp;amp; 0 \\ 0 &amp;amp; 0 &amp;amp; 1&lt;br /&gt;
  \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
It is important to note that ''a matrix is invertible if and only if its determinant is nonzero.''  Thus one only needs to calculate the&lt;br /&gt;
determinant to see if a matrix has an inverse or not.&lt;br /&gt;
&lt;br /&gt;
====Hermitian Matrices====&lt;br /&gt;
&lt;br /&gt;
Hermitian matrices are important for a variety of reasons; primarily, it is because their eigenvalues are real.  Thus Hermitian matrices are used to represent density operators and density matrices, as well as Hamiltonians.  The density operator is a positive semi-definite Hermitian matrix (it has no negative eigenvalues) that has its trace equal to one.  In any case, it is often desirable to represent &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrices using a real linear combination of a complete set of &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrices.  A set of &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrices is complete if any Hermitian matrix can be represented in terms of the set.  Let &amp;lt;math&amp;gt;\{\lambda_i\}\,\!&amp;lt;/math&amp;gt; be a complete set.  Then any Hermitian matrix can be represented by &amp;lt;math&amp;gt;\sum_i a_i \lambda_i\,\!&amp;lt;/math&amp;gt;.  The set can always be taken to be a set of traceless Hermitian matrices and the identity matrix.  This is convenient for the density matrix (its trace is one) because the identity part of an &amp;lt;math&amp;gt;N\times N\,\!&amp;lt;/math&amp;gt; Hermitian matrix is &amp;lt;math&amp;gt;(1/N)\mathbb{I}\,\!&amp;lt;/math&amp;gt; if we take all others in the set to be traceless.  For the Hamiltonian, the set consists of a traceless part and an identity part where identity part just gives an overall phase which can often be neglected.  &lt;br /&gt;
&lt;br /&gt;
One example of such a set which is extremely useful is the set of Pauli matrices.  These are discussed in detail in [[Chapter 2 - Qubits and Collections of Qubits|Chapter 2]] and in particular in [[Chapter 2 - Qubits and Collections of Qubits#The Pauli Matrices|Section 2.4]].&lt;br /&gt;
&lt;br /&gt;
====Unitary Matrices====&lt;br /&gt;
&lt;br /&gt;
A ''unitary matrix'' &amp;lt;!-- \index{unitary matrix} --&amp;gt; &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; is one whose&lt;br /&gt;
inverse is also its Hermitian conjugate, &amp;lt;math&amp;gt;U^\dagger = U^{-1}\,\!&amp;lt;/math&amp;gt;, so that &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
U^\dagger U = UU^\dagger = \mathbb{I}.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If the unitary matrix also has determinant one, it is said to be ''a special unitary matrix''.&amp;lt;!-- \index{special unitary matrix}--&amp;gt;  The set of&lt;br /&gt;
&amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; unitary matrices is denoted&lt;br /&gt;
&amp;lt;math&amp;gt;U(n)\,\!&amp;lt;/math&amp;gt; and the set of special unitary matrices is denoted &amp;lt;math&amp;gt;SU(n)\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
Unitary matrices are particularly important in quantum mechanics&lt;br /&gt;
because they describe the evolution of quantum states.&lt;br /&gt;
They have this ability due to the fact that the rows and columns of unitary matrices (viewed as vectors) are orthonormal. (This is made clear in an example below.)  This means that when&lt;br /&gt;
they act on a basis vector of the form&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt; \left\vert j\right\rangle = &lt;br /&gt;
 \left(\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 &lt;br /&gt;
  \end{array}\right), &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.11}}&lt;br /&gt;
with a single 1, in say the &amp;lt;math&amp;gt;j&amp;lt;/math&amp;gt;th spot, and zeroes everywhere else, the result is a normalized complex vector.  Acting on a set of&lt;br /&gt;
orthonormal vectors of the form given in Eq.[[#eqC.11|(C.11)]]&lt;br /&gt;
will produce another orthonormal set.  &lt;br /&gt;
&lt;br /&gt;
Let us consider the example of a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; unitary matrix, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
U = \left(\begin{array}{cc} &lt;br /&gt;
              a &amp;amp; b \\ &lt;br /&gt;
              c &amp;amp; d &lt;br /&gt;
           \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.12}}&lt;br /&gt;
The inverse of this matrix is the Hermitian conjugate, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
U ^{-1} = U^\dagger = \left(\begin{array}{cc} &lt;br /&gt;
                         a^* &amp;amp; c^* \\ &lt;br /&gt;
                         b^* &amp;amp; d^* &lt;br /&gt;
                       \end{array}\right),&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.13}}&lt;br /&gt;
provided that the matrix &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; satisfies the constraints&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
|a|^2 + |b|^2 = 1, \; &amp;amp; \; ac^*+bd^* =0  \\&lt;br /&gt;
ca^*+db^*=0,  \;      &amp;amp;  \; |c|^2 + |d|^2 =1,&lt;br /&gt;
\end{align}\,\!&amp;lt;/math&amp;gt;|C.14}}&lt;br /&gt;
and&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
|a|^2 + |c|^2 = 1, \; &amp;amp; \; ba^*+dc^* =0  \\&lt;br /&gt;
b^*a+d^*c=0,  \;      &amp;amp;  \; |b|^2 + |d|^2 =1.&lt;br /&gt;
\end{align}\,\!&amp;lt;/math&amp;gt;|C.15}}&lt;br /&gt;
Looking at each row as a vector, the constraints in&lt;br /&gt;
Eq.[[#eqC.14|(C.14)]] are the orthonormality conditions for the&lt;br /&gt;
vectors forming the rows.  Similarly, the constraints in&lt;br /&gt;
Eq.[[#eqC.15|(C.15)]] are the orthonormality conditions for the&lt;br /&gt;
vectors forming the columns.&lt;br /&gt;
&lt;br /&gt;
===More Dirac Notation===&lt;br /&gt;
&lt;br /&gt;
Now that we have a definition for the Hermitian conjugate, we consider the&lt;br /&gt;
case for a &amp;lt;math&amp;gt;1\times n\,\!&amp;lt;/math&amp;gt; matrix, i.e. a vector.  In Dirac notation, this is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The Hermitian conjugate comes up so often that we use the following&lt;br /&gt;
notation for vectors:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle \psi\right\vert = (\left\vert\psi\right\rangle)^\dagger = \left(\begin{array}{c} \alpha \\&lt;br /&gt;
    \beta \end{array}\right)^\dagger &lt;br /&gt;
 = \left( \alpha^*, \; \beta^* \right).  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This is a row vector and in Dirac notation is denoted by the symbol &amp;lt;math&amp;gt;\left\langle\cdot \right\vert\!&amp;lt;/math&amp;gt;, which is called a ''bra''.  Let us consider a second complex vector, &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta&lt;br /&gt;
  \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The ''inner product'' &amp;lt;!-- \index{inner product}--&amp;gt; between &amp;lt;math&amp;gt;\left\vert\psi\right\rangle\,\!&amp;lt;/math&amp;gt; and &lt;br /&gt;
&amp;lt;math&amp;gt;\left\vert\phi\right\rangle\,\!&amp;lt;/math&amp;gt; is computed as follows:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
 \left\langle\phi\mid\psi\right\rangle &amp;amp; \equiv (\left\vert\phi\right\rangle)^\dagger\left\vert\psi \right\rangle   \\&lt;br /&gt;
                  &amp;amp;= (\gamma^*,\delta^*) \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right)   \\&lt;br /&gt;
                  &amp;amp;= \gamma^*\alpha + \delta^*\beta.&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.16}}&lt;br /&gt;
If these two vectors are ''orthogonal'', &amp;lt;!-- \index{orthogonal!vectors} --&amp;gt;&lt;br /&gt;
then their inner product is zero, or &amp;lt;math&amp;gt;\left\langle\phi\mid\psi\right\rangle =0\,\!&amp;lt;/math&amp;gt;.  (The &amp;lt;math&amp;gt; \left\langle\phi\mid\psi\right\rangle \,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
is called a ''bracket'', which is the product of the ''bra'' and the ''ket''.)  The inner product of &amp;lt;math&amp;gt;\left\vert\psi\right\rangle\,\!&amp;lt;/math&amp;gt; with itself is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle\psi\mid\psi\right\rangle = |\alpha|^2 + |\beta|^2.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This vector is considered normalized when &amp;lt;math&amp;gt;\left\langle\psi\mid\psi\right\rangle = 1\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
More generally, we will consider vectors in &amp;lt;math&amp;gt;N\,\!&amp;lt;/math&amp;gt; dimensions.  In this&lt;br /&gt;
case we write the vector in terms of a set of basis vectors,&lt;br /&gt;
&amp;lt;math&amp;gt;\{\left\vert i\right\rangle\}\,\!&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;i = 0,1,2,...N-1\,\!&amp;lt;/math&amp;gt;.  This is an ordered set of&lt;br /&gt;
vectors which are labeled simply by integers.  If the set is orthogonal,&lt;br /&gt;
then &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i\mid j\right\rangle = 0, \;\; \mbox{for all }i\neq j.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If they are normalized, then &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i \mid i \right\rangle = 1, \;\;\mbox{for all } i.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
If both of these are true, i.e. the entire set is orthonormal, we can&lt;br /&gt;
write&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i\mid j\right\rangle = \delta_{ij}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where the symbol &amp;lt;math&amp;gt;\delta_{ij}\,\!&amp;lt;/math&amp;gt; is called the Kronecker delta &amp;lt;!-- \index{Kronecker delta} --&amp;gt; and is defined by &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\delta_{ij} = \begin{cases}&lt;br /&gt;
               1, &amp;amp; \mbox{if } i=j, \\&lt;br /&gt;
               0, &amp;amp; \mbox{if } i\neq j.&lt;br /&gt;
              \end{cases}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.17}}&lt;br /&gt;
Now consider &amp;lt;math&amp;gt;(N+1)\,\!&amp;lt;/math&amp;gt;-dimensional vectors by letting two such vectors&lt;br /&gt;
be expressed in the same basis as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert \Psi\right\rangle = \sum_{i=0}^{N} \alpha_i\left\vert i\right\rangle&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
and&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\Phi\right\rangle = \sum_{j=0}^{N} \beta_j\left\vert j\right\rangle.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Then the inner product &amp;lt;!--\index{inner product}--&amp;gt; is&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\langle\Psi\mid\Phi\right\rangle &amp;amp;= \left(\sum_{i=0}^{N}&lt;br /&gt;
             \alpha_i\left\vert i\right\rangle\right)^\dagger\left(\sum_{j=0}^{N} \beta_j\left\vert j\right\rangle\right)  \\&lt;br /&gt;
                 &amp;amp;= \sum_{ij} \alpha_i^*\beta_j\left\langle i\mid j\right\rangle  \\&lt;br /&gt;
                 &amp;amp;= \sum_{ij} \alpha_i^*\beta_j\delta_{ij}  \\&lt;br /&gt;
                 &amp;amp;= \sum_i\alpha^*_i\beta_i,&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.18}}&lt;br /&gt;
where the fact that the delta function is zero unless&lt;br /&gt;
&amp;lt;math&amp;gt;i=j\,\!&amp;lt;/math&amp;gt; is used to obtain the last equality.  Taking the inner product of a vector&lt;br /&gt;
with itself will get&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle\Psi\mid\Psi\right\rangle = \sum_i\alpha^*_i\alpha_i = \sum_i|\alpha_i|^2.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This immediately gives us a very important property of the inner&lt;br /&gt;
product.  It tells us that, in general,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle\Phi\mid\Phi\right\rangle \geq 0, \;\; \mbox{and} \;\; \left\langle\Phi\mid \Phi\right\rangle = 0&lt;br /&gt;
\Leftrightarrow \left\vert\Phi\right\rangle = 0. &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
(The symbol &amp;lt;math&amp;gt;\Leftrightarrow\,\!&amp;lt;/math&amp;gt; means &amp;lt;nowiki&amp;gt;&amp;quot;if and only if,&amp;quot;&amp;lt;/nowiki&amp;gt; sometimes written as &amp;lt;nowiki&amp;gt;&amp;quot;iff.&amp;quot;&amp;lt;/nowiki&amp;gt;)  &lt;br /&gt;
&lt;br /&gt;
We could also expand a vector in a different basis.  Let us suppose&lt;br /&gt;
that the set &amp;lt;math&amp;gt;\{\left\vert e_k \right\rangle\}\,\!&amp;lt;/math&amp;gt; is an orthonormal basis &amp;lt;math&amp;gt;(\left\langle e_k \mid e_l\right\rangle =&lt;br /&gt;
\delta_{kl})\,\!&amp;lt;/math&amp;gt; that is different from the one considered earlier.  We&lt;br /&gt;
could expand our vector &amp;lt;math&amp;gt;\left\vert\Psi\right\rangle\,\!&amp;lt;/math&amp;gt; in terms of our new basis by&lt;br /&gt;
expanding our new basis in terms of our old basis.  Let us first&lt;br /&gt;
expand the &amp;lt;math&amp;gt;\left\vert e_k\right\rangle\,\!&amp;lt;/math&amp;gt; in terms of the &amp;lt;math&amp;gt;\left\vert j\right\rangle\,\!&amp;lt;/math&amp;gt;:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert e_k\right\rangle= \sum_j \left\vert j\right\rangle \left\langle j\mid e_k\right\rangle,&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.19}}&lt;br /&gt;
so that &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert \Psi\right\rangle &amp;amp;= \sum_j \alpha_j\left\vert j\right\rangle  \\&lt;br /&gt;
           &amp;amp;= \sum_{j}\sum_k\alpha_j\left\vert e_k \right\rangle \left\langle e_k \mid j\right\rangle  \\ &lt;br /&gt;
           &amp;amp;= \sum_k \alpha_k^\prime \left\vert e_k\right\rangle, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.20}}&lt;br /&gt;
where &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j\right\rangle. &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.21}}&lt;br /&gt;
Notice that the insertion of &amp;lt;math&amp;gt;\sum_k\left\vert e_k\right\rangle\left\langle e_k\right\vert\,\!&amp;lt;/math&amp;gt; didn't do anything to our original vector; it is the same vector, just in a&lt;br /&gt;
different basis.  Therefore, this is effectively the identity operator,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\mathbb{I} = \sum_k\left\vert e_k \right\rangle\left\langle e_k\right\vert.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This is an important and quite useful relation.  &lt;br /&gt;
To interpret Eq.[[#eqC.19|(C.19)]], we can draw a close&lt;br /&gt;
analogy with three-dimensional real vectors.  The inner product&lt;br /&gt;
&amp;lt;math&amp;gt;\left\vert e_k \right\rangle \left\langle j \right\vert\,\!&amp;lt;/math&amp;gt; can be interpreted as the projection of one vector onto&lt;br /&gt;
another.  This provides the part of &amp;lt;math&amp;gt;\left\vert j \right\rangle\,\!&amp;lt;/math&amp;gt; along &amp;lt;math&amp;gt;\left\vert e_k \right\rangle\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
===Transformations===&lt;br /&gt;
&lt;br /&gt;
Suppose we have two different orthogonal bases, &amp;lt;math&amp;gt;\{e_k\}\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;\{j\}\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
The numbers &amp;lt;math&amp;gt;\left\langle e_k\mid j\right\rangle\,\!&amp;lt;/math&amp;gt; for all the different &amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;j\,\!&amp;lt;/math&amp;gt; are&lt;br /&gt;
often referred to as matrix elements since the set forms a matrix, with&lt;br /&gt;
&amp;lt;math&amp;gt;k\,\!&amp;lt;/math&amp;gt; labelling the rows and &amp;lt;math&amp;gt;j\,\!&amp;lt;/math&amp;gt; labelling the columns.  Thus we&lt;br /&gt;
can represent the transformation from one basis to another with a matrix&lt;br /&gt;
transformation.  Let &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; be the matrix with elements &amp;lt;math&amp;gt;m_{kj} =&lt;br /&gt;
\left\langle e_k\mid j\right\rangle\,\!&amp;lt;/math&amp;gt;.  The transformation from one basis to another,&lt;br /&gt;
written in terms of the coefficients of &amp;lt;math&amp;gt;\left\vert\Psi\right\rangle\,\!&amp;lt;/math&amp;gt;, is &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt; A^\prime = MA, &amp;lt;/math&amp;gt;|C.22}}&lt;br /&gt;
where &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
A^\prime = \left(\begin{array}{c} \alpha_1^\prime \\ \alpha_2^\prime \\ \vdots \\&lt;br /&gt;
    \alpha_n^\prime \end{array}\right), \;\; &lt;br /&gt;
\mbox{ and } \;\;&lt;br /&gt;
A = \left(\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\&lt;br /&gt;
    \alpha_n\end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This sort of transformation is a change of basis.  Oftentimes when one vector is transformed to another the transformation can be viewed as a transformation of the components of the vector and is also represented by a matrix.  Thus transformations can either be&lt;br /&gt;
represented by the matrix equation, like Eq.[[#eqC.22|(C.22)]], or the components, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\alpha_k^\prime = \sum_j \alpha_j \left\langle e_k \mid j \right\rangle = \sum_j m_{kj}\alpha_j. &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.23}}&lt;br /&gt;
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.  &lt;br /&gt;
&lt;br /&gt;
For a general transformation matrix &amp;lt;math&amp;gt;T\,\!&amp;lt;/math&amp;gt; acting on a vector,&lt;br /&gt;
the matrix elements in a particular basis &amp;lt;math&amp;gt;\left\vert i\right\rangle\,\!&amp;lt;/math&amp;gt; are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
t_{ij} = \left\langle i\right\vert (T) \left\vert j\right\rangle, &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
just as elements of a vector can be found using&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\langle i\mid \Psi \right\rangle = \alpha_i.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Transformations of a Qubit====&lt;br /&gt;
&lt;br /&gt;
It is worth belaboring the point somewhat and presenting several ways in which to parametrize the set of transformations of a qubit.  A qubit state is represented by a complex two-dimensional vector that has been normalized to one:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert \psi\right\rangle = \alpha_0 \left\vert 0 \right\rangle + \alpha_1 \left\vert 1\right\rangle = \left(\begin{array}{c} \alpha_0 \\ \alpha_1 \end{array}\right), \;\;\;\; |\alpha_0|^2 + |\alpha_1|^2 = 1.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The most general matrix transformation that will take this to any other state of the same form (complex, 2-d vector with unit norm) is a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; unitary matrix.  In [[Chapter 2 - Qubits and Collections of Qubits|Chapter 2]], several specific examples of qubit transformations were given; in [[Chapter 3 - Physics of Quantum Information|Chapter 3]], [[Chapter 3 - Physics of Quantum Information#Measurements Revisited|Section 3.4]] it was stated that an element of SU(2) can be written as (see [[Chapter 3 - Physics of Quantum Information#Exponentian of a Matrix|Section 3.2.1, Exponentiation of a Matrix]], in particular [[Chapter 3 - Physics of Quantum Information#eq3.8|Eq. (3.8)]])&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
U(\theta) &amp;amp;= \exp(-i\vec{n}\cdot\vec{\sigma} \theta/2) \\&lt;br /&gt;
          &amp;amp;= (\mathbb{I}\cos(\theta/2) -i\vec{n}\cdot\vec{\sigma} \sin(\theta/2))&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.24}}&lt;br /&gt;
where &amp;lt;math&amp;gt;\vec{n}\,\!&amp;lt;/math&amp;gt; is a unit vector, &amp;lt;math&amp;gt;|\vec{n}|=1\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;\vec{n}\cdot\vec{\sigma} =&lt;br /&gt;
n_1\sigma_1+n_2\sigma_2+n_3\sigma_3\,\!&amp;lt;/math&amp;gt;. &lt;br /&gt;
Explicitly, this is &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
 \exp(-i\vec{n}\cdot\vec{\sigma} \theta/2) &amp;amp;= \left(\begin{array}{cc}&lt;br /&gt;
                                  1 &amp;amp; 0 \\ &lt;br /&gt;
                                  0 &amp;amp; 1 \end{array}\right)\cos(\theta/2) \\&lt;br /&gt;
                        &amp;amp; \;\;\;   + (-i)\left[ n_1\left(\begin{array}{cc}&lt;br /&gt;
                                  0 &amp;amp; 1 \\ &lt;br /&gt;
                                  1 &amp;amp; 0 \end{array}\right)&lt;br /&gt;
                              + n_2\left(\begin{array}{cc}&lt;br /&gt;
                                  0 &amp;amp; -i \\ &lt;br /&gt;
                                  i &amp;amp; 0 \end{array}\right)&lt;br /&gt;
                              + n_3\left(\begin{array}{cc}&lt;br /&gt;
                                  1 &amp;amp; 0 \\ &lt;br /&gt;
                                  0 &amp;amp; -1 \end{array}\right)\right]\sin(\theta/2) \\&lt;br /&gt;
                                &amp;amp;= &lt;br /&gt;
         \left(\begin{array}{cc}&lt;br /&gt;
  \cos(\theta/2) -in_3\sin(\theta/2) &amp;amp; (-in_1-n_2)\sin(\theta/2) \\ &lt;br /&gt;
   (-in_1+n_2)\sin(\theta/2) &amp;amp; \cos(\theta/2) +in_3\sin(\theta/2)  \end{array}\right).&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Let us prove this.  First, using [[Chapter 3 - Physics of Quantum Information#eq3.7|Eq. (3.7)]], we will need to find &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})^m\,\!&amp;lt;/math&amp;gt;, i.e., all powers of &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})\,\!&amp;lt;/math&amp;gt;.  This turns out to be fairly easy.  First note that &amp;lt;math&amp;gt;\hat{n}\cdot \hat{n} = 1\,\!&amp;lt;/math&amp;gt; since &amp;lt;math&amp;gt;\hat{n}\,\!&amp;lt;/math&amp;gt; is a unit vector.  Then note that &lt;br /&gt;
&amp;lt;math&amp;gt;\sigma_i\sigma_j = \mathbb{I}\delta_{ij} +i \epsilon_{ijk}\sigma_k,&amp;lt;/math&amp;gt;.  (See [[Chapter 2 - Qubits and Collections of Qubits#eq2.21|Eq. (2.21)]] as well as Eqs. [[Appendix C - Vectors and Linear Algebra#eqC.17|(C.17)]] and [[Appendix C - Vectors and Linear Algebra#eqC.8|(C.8)]].)  These imply that (recalling that &amp;lt;math&amp;gt; \sigma_x =\sigma_1,\; \sigma_y = \sigma_2, \;\;\sigma_z=\sigma_3 &amp;lt;/math&amp;gt;),&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
(\vec{n}\cdot\vec{\sigma})^2 &amp;amp;= (n_1 \sigma_1 + n_2 \sigma_2 + n_3 \sigma_3)^2 \\&lt;br /&gt;
                  &amp;amp;= \left( \sum_i n_i \sigma_i\right) \left( \sum_j n_j \sigma_j \right) \\&lt;br /&gt;
                  &amp;amp;= \sum_{ij} n_i n_j \sigma_i \sigma_j \\&lt;br /&gt;
                  &amp;amp;= \sum_{ij} n_i n_j (\mathbb{I} \delta_{ij}+i\epsilon_{ijk}\sigma_k ) \\&lt;br /&gt;
                  &amp;amp;= \sum_{ij} n_i n_i   \mathbb{I} = \mathbb{I}.&lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
To get the third line, one just uses [[Chapter 2 - Qubits and Collections of Qubits#eq2.21|Eq. (2.21)]].  To see that &amp;lt;math&amp;gt;\sum_{ij} n_i n_j \epsilon_{ijk}\sigma_k \,\!&amp;lt;/math&amp;gt; is zero, note that &amp;lt;math&amp;gt;n_i n_j  \,\!&amp;lt;/math&amp;gt; is symmetric in &amp;lt;math&amp;gt;i,j\,\!&amp;lt;/math&amp;gt; but &amp;lt;math&amp;gt;\epsilon_{ijk} \,\!&amp;lt;/math&amp;gt; is antisymmetric in &amp;lt;math&amp;gt;i,j\,\!&amp;lt;/math&amp;gt;.  Another way to see this more explicitly is to write out the sum and notice that each term shows up with a + and - sign thus cancelling each other out.  Therefore, all the even powers of &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})\,\!&amp;lt;/math&amp;gt; are just equal to the identity matrix and all odd powers are just &amp;lt;math&amp;gt;(\hat{n}\cdot \vec{\sigma})\,\!&amp;lt;/math&amp;gt; times the even parts.  Thus the sum in [[Chapter 3 - Physics of Quantum Information#eq3.7|Eq. (3.7)]] reduces to &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\exp(-i \hat{n}\cdot \vec{\sigma}\theta/2) &amp;amp; = \mathbb{I} \cos(\theta/2) + i (\hat{n}\cdot \vec{\sigma})\sin(\theta/2). \;\;\;\; \square&lt;br /&gt;
\end{align}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Notice that this is a ''special unitary matrix.''  (See Section [[Appendix C - Vectors and Linear Algebra#Unitary Matrices|Unitary Matrices]].)&lt;br /&gt;
To see that this is the most general SU(2) matrix, one needs to verify that any complex &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; unitary matrix can be written in this form.  (One way to do this is to start with a generic matrix and impose the restrictions.  Here one may simply convince oneself that this is general through observation by acting on basis vectors.)  This is the most general qubit transformation and can be interpreted as a rotation about the axis &amp;lt;math&amp;gt;\hat{n}\,\!&amp;lt;/math&amp;gt; by an angle &amp;lt;math&amp;gt;\theta\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
&lt;br /&gt;
Another parametrization of this set of matrices is the following, called the Euler angle parametrization:&lt;br /&gt;
 {{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
U_{EA}   = \exp(-i\sigma_z \alpha/2) \exp(-i\sigma_y \beta/2) \exp(-i\sigma_z \gamma/2).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.25}}&lt;br /&gt;
In this case the matrices &amp;lt;math&amp;gt;\sigma_z\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\sigma_y\,\!&amp;lt;/math&amp;gt; are not unique.  Any two of the three Pauli matrices (or one of each) may be chosen.  This is quite simple, useful, and generalizable to SU(N) for N arbitrary.  In the simple case of a qubit, one may convince oneself by acting on basis vectors as before.  However, with a little thought, one may see that rotating to a position on the sphere by the first angle, followed by rotations using the other two, will provide for a general orientation of an object.&lt;br /&gt;
&lt;br /&gt;
====Similarity Transformation====&lt;br /&gt;
&lt;br /&gt;
A ''similarity transformation'' &amp;lt;!--\index{similarity transformation}--&amp;gt; &lt;br /&gt;
of an &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; matrix &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; by an invertible matrix &amp;lt;math&amp;gt;S\,\!&amp;lt;/math&amp;gt; is &amp;lt;math&amp;gt;S A S^{-1}\,\!&amp;lt;/math&amp;gt;.&lt;br /&gt;
There are (at least) two important things to note about similarity&lt;br /&gt;
transformations: &lt;br /&gt;
#Similarity transformations leave the trace of a matrix unchanged.  This is shown explicitly in [[#The Trace|Section 3.5]].&lt;br /&gt;
#Similarity transformations leave the determinant of a matrix unchanged, or invariant.  This is because &amp;lt;center&amp;gt;&amp;lt;math&amp;gt; \det(SAS^{-1}) = \det(S)\det(A)\det(S^{-1}) =\det(S)\det(A)\frac{1}{\det(S)} = \det(A). \,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
#Simultaneous similarity transformations of matrices in an equation will leave the equation unchanged.  Let &amp;lt;math&amp;gt;A^\prime = SAS^{-1}\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;B^\prime = SBS^{-1}\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;C^\prime = SCS^{-1}\,\!&amp;lt;/math&amp;gt;.  If &amp;lt;math&amp;gt;AB=C\,\!&amp;lt;/math&amp;gt;, then &amp;lt;math&amp;gt;A^\prime B^\prime = C^\prime\,\!&amp;lt;/math&amp;gt;, since &amp;lt;math&amp;gt;A^\prime B^\prime = SAS^{-1}SBS^{-1} = SABS^{-1} =  SCS^{-1}=C^\prime\,\!&amp;lt;/math&amp;gt;.  The two matrices &amp;lt;math&amp;gt;A^\prime\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; are said to be ''similar''.&lt;br /&gt;
&amp;lt;!-- \index{similar matrices} --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Eigenvalues and Eigenvectors===&lt;br /&gt;
&lt;br /&gt;
&amp;lt;!-- \index{eigenvalues}\index{eigenvectors} --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A matrix can always be diagonalized.  By this, it is meant that for&lt;br /&gt;
every complex matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; there is a diagonal matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; such that &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
M = UDV,  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.26}}&lt;br /&gt;
where &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;V\,\!&amp;lt;/math&amp;gt; are unitary matrices.  This form is called a singular value decomposition of the matrix and the entries of the diagonal matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; are called the ''singular values'' &amp;lt;!--\index{singular values}--&amp;gt; &lt;br /&gt;
of the matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt;.  However, the singular values are not always easy to find.  &lt;br /&gt;
&lt;br /&gt;
For the special case that the matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; is Hermitian &amp;lt;math&amp;gt;(M^\dagger = M)\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
the matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt; can be written as&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
M = U D U^\dagger&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.27}}&lt;br /&gt;
where &amp;lt;math&amp;gt;U\,\!&amp;lt;/math&amp;gt; is unitary &amp;lt;math&amp;gt;(U^{-1}=U^\dagger)\,\!&amp;lt;/math&amp;gt;.  In this case the elements&lt;br /&gt;
of the matrix &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; are called ''eigenvalues''. &amp;lt;!--\index{eigenvalues}--&amp;gt;&lt;br /&gt;
Very often eigenvalues are introduced as solutions to the equation&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
M \left\vert v\right\rangle = \lambda \left\vert v\right\rangle&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where &amp;lt;math&amp;gt;\left\vert v\right\rangle\,\!&amp;lt;/math&amp;gt; is an ''eigenvector''. &amp;lt;!--\index{eigenvector} --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
To find the eigenvalues and eigenvectors of a matrix &amp;lt;math&amp;gt;M\,\!&amp;lt;/math&amp;gt;, we follow a&lt;br /&gt;
standard procedure which is to calculate&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(\lambda\mathbb{I} - M) = 0&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.28}}&lt;br /&gt;
and then solve for &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  The different solutions for &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt; is the&lt;br /&gt;
set of eigenvalues and is called the ''spectrum''. &amp;lt;!-- \index{spectrum}--&amp;gt; Let the different eigenvalues be denoted by &amp;lt;math&amp;gt;\lambda_i\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;i=1,2,...,n\,\!&amp;lt;/math&amp;gt; fo an &amp;lt;math&amp;gt;n\times n\,\!&amp;lt;/math&amp;gt; vector.  If two&lt;br /&gt;
eigenvalues are equal, we say the spectrum is &lt;br /&gt;
''degenerate''. &amp;lt;!--\index{degenerate}--&amp;gt; To find the&lt;br /&gt;
eigenvectors, which correspond to different eigenvalues, the equation &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
M \left\vert v\right\rangle = \lambda_i \left\vert v\right\rangle&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
must be solved for each value of &amp;lt;math&amp;gt;i\,\!&amp;lt;/math&amp;gt;.  Notice that this equation&lt;br /&gt;
holds even if we multiply both sides by some complex number.  This&lt;br /&gt;
implies that an eigenvector can always be scaled.  Usually they are&lt;br /&gt;
normalized to obtain an orthonormal set.  As we will see by example,&lt;br /&gt;
degenerate eigenvalues require some care.  &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
====Example 1====&lt;br /&gt;
&lt;br /&gt;
Consider a &amp;lt;math&amp;gt;2\times 2\,\!&amp;lt;/math&amp;gt; Hermitian matrix&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\sigma = \left(\begin{array}{cc} &lt;br /&gt;
               1+a &amp;amp; b-ic \\&lt;br /&gt;
              b+ic &amp;amp; 1-a  \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.29}}&lt;br /&gt;
To find the eigenvalues &amp;lt;!--\index{eigenvalues}--&amp;gt; &lt;br /&gt;
of this, we follow a standard procedure, which&lt;br /&gt;
is to calculate &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
\det(\sigma-\lambda\mathbb{I}) = 0,&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.30}}&lt;br /&gt;
and solve for &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  The eigenvalues of this matrix are given by&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det\left(\begin{array}{cc} &lt;br /&gt;
               1+a-\lambda &amp;amp; b-ic \\&lt;br /&gt;
              b+ic &amp;amp; 1-a-\lambda  \end{array}\right) =0,  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
which implies that the eigenvalues are&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\lambda_{\pm} = 1\pm \sqrt{a^2+b^2+c^2}&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
and the eigenvectors &amp;lt;!--\index{eigenvectors}--&amp;gt; are&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_1=\left(\begin{array}{c}&lt;br /&gt;
        i\left(-a + c + \sqrt{a^2 + 4 b^2 - 2 ac + c^2} \right) \\ &lt;br /&gt;
        2b &lt;br /&gt;
        \end{array}\right), &lt;br /&gt;
v_2= \left(\begin{array}{c}&lt;br /&gt;
         i\left(-a + c - \sqrt {a^2 + 4 b^2 - 2 a c + c^2} \right)\\ &lt;br /&gt;
         2b \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
These expressions are useful for calculating properties of qubit&lt;br /&gt;
states as will be seen in the text.&lt;br /&gt;
&lt;br /&gt;
====Example 2====&lt;br /&gt;
&lt;br /&gt;
Now consider a &amp;lt;math&amp;gt;3\times 3\,\!&amp;lt;/math&amp;gt; matrix,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
N= \left(\begin{array}{ccc}&lt;br /&gt;
              1 &amp;amp; -i &amp;amp; 0 \\&lt;br /&gt;
              i &amp;amp; 1 &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 0 &amp;amp; 1 \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
First we calculate&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det\left(\begin{array}{ccc}&lt;br /&gt;
              1-\lambda &amp;amp; -i &amp;amp; 0 \\&lt;br /&gt;
              i         &amp;amp; 1-\lambda  &amp;amp; 0 \\&lt;br /&gt;
              0         &amp;amp;       0    &amp;amp; 1-\lambda &lt;br /&gt;
           \end{array}\right) &lt;br /&gt;
    = (1-\lambda)[(1-\lambda)^2-1].&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This implies that the eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']] are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\lambda = 1,0, \mbox{ or } 2.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Let &amp;lt;math&amp;gt;\lambda_1=1\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;\lambda_0 = 0\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;\lambda_2 = 2\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
To find eigenvectors, &amp;lt;!--\index{eigenvalues}--&amp;gt; we calculate&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
Nv &amp;amp;= \lambda v, \\&lt;br /&gt;
\left(\begin{array}{ccc}&lt;br /&gt;
              1 &amp;amp; -i &amp;amp; 0 \\&lt;br /&gt;
              i &amp;amp; 1 &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 0 &amp;amp; 1 \end{array}\right)\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right) &amp;amp;= \lambda\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right)&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.31}}&lt;br /&gt;
for each &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
For &amp;lt;math&amp;gt;\lambda = 1\,\!&amp;lt;/math&amp;gt;, we get the following equations:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_1 -iv_2 &amp;amp;= v_1, \\&lt;br /&gt;
iv_1+v_2 &amp;amp;= v_2,  \\&lt;br /&gt;
v_3 &amp;amp;= v_3. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.32}}&lt;br /&gt;
Solving this obtains &amp;lt;math&amp;gt;v_2 =0\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;v_1 =0\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;v_3\,\!&amp;lt;/math&amp;gt; is any non-zero number (which will be chosen to normalize the vector).  For &amp;lt;math&amp;gt;\lambda&lt;br /&gt;
=0\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_1  &amp;amp;= iv_2, \\&lt;br /&gt;
v_3 &amp;amp;= 0. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.33}}&lt;br /&gt;
And finally, for &amp;lt;math&amp;gt;\lambda = 2\,\!&amp;lt;/math&amp;gt;, we obtain&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_1 -iv_2 &amp;amp;= 2v_1, \\&lt;br /&gt;
iv_1+v_2 &amp;amp;= 2v_2, \\&lt;br /&gt;
v_3 &amp;amp;= 2v_3, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.34}}&lt;br /&gt;
so that &amp;lt;math&amp;gt;v_1 = -iv_2\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
Therefore, our three eigenvectors &amp;lt;!--\index{eigenvalues}--&amp;gt; are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_0 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 1\\ 0 \end{array}\right), \; &lt;br /&gt;
v_1 = \left(\begin{array}{c} 0 \\ 0\\ 1 \end{array}\right), \; &lt;br /&gt;
v_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} -i \\ 1\\ 0 \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
The matrix &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
V= (v_0,v_1,v_2) = \left(\begin{array}{ccc}&lt;br /&gt;
              i/\sqrt{2} &amp;amp; 0     &amp;amp; -i/\sqrt{2} \\&lt;br /&gt;
              1/\sqrt{2} &amp;amp; 0     &amp;amp; 1/\sqrt{2} \\&lt;br /&gt;
              0          &amp;amp; 1     &amp;amp; 0 \end{array}\right)&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
is the matrix that diagonalizes &amp;lt;math&amp;gt;N\,\!&amp;lt;/math&amp;gt; in the following way:&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
N = VDV^\dagger&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
where&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
D = \left(\begin{array}{ccc}&lt;br /&gt;
              0 &amp;amp; 0  &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 1  &amp;amp; 0 \\&lt;br /&gt;
              0 &amp;amp; 0  &amp;amp; 2 \end{array}\right)&lt;br /&gt;
.\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
We may write this as&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
V^\dagger N V = D.   &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This is sometimes called the ''eigenvalue decompostion''&amp;lt;!--\index{eigenvalue decomposition}--&amp;gt;  of the matrix and can also be written as&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
N = \sum_i \lambda_i v_iv^\dagger_i.  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;|C.35}}&lt;br /&gt;
&lt;br /&gt;
====Example 3====&lt;br /&gt;
&lt;br /&gt;
Next, consider the complex &amp;lt;math&amp;gt;3\times 3&amp;lt;/math&amp;gt; Hermitian matrix &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
M = \left(\begin{array}{ccc}&lt;br /&gt;
              \frac{5}{2} &amp;amp; 0  &amp;amp; \frac{i}{2} \\&lt;br /&gt;
              0 &amp;amp; 2  &amp;amp; 0 \\&lt;br /&gt;
              -\frac{i}{2} &amp;amp; 0  &amp;amp; \frac{5}{2} \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
First we calculate&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\det\left(\begin{array}{ccc}&lt;br /&gt;
              \frac{5}{2}-\lambda &amp;amp; 0 &amp;amp; \frac{i}{2} \\&lt;br /&gt;
              0         &amp;amp; 2-\lambda  &amp;amp; 0 \\&lt;br /&gt;
              -\frac{i}{2}         &amp;amp;       0    &amp;amp; \frac{5}{2}-\lambda &lt;br /&gt;
           \end{array}\right) &lt;br /&gt;
    = (2-\lambda)\left[\left(\frac{5}{2}-\lambda\right)^2-\frac{1}{4}\right].&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
This implies that the eigenvalues [[Appendix C - Vectors and Linear Algebra#Eigenvalues and Eigenvectors|'''C.6''']] are &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\lambda = 2,2, \mbox{ or } 3.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Note that there are two that are the same, or degenerate.  &lt;br /&gt;
Let &amp;lt;math&amp;gt;\lambda_1=2\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;\lambda_2 = 2\,\!&amp;lt;/math&amp;gt;, and &amp;lt;math&amp;gt;\lambda_3 = 3\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
To find eigenvectors, &amp;lt;!--\index{eigenvalues}--&amp;gt; we calculate&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
Mv &amp;amp;= \lambda v, \\&lt;br /&gt;
\left(\begin{array}{ccc}&lt;br /&gt;
              \frac{5}{2} &amp;amp; 0 &amp;amp; \frac{i}{2} \\&lt;br /&gt;
              0 &amp;amp; 2 &amp;amp; 0 \\&lt;br /&gt;
              -\frac{i}{2} &amp;amp; 0 &amp;amp; \frac{5}{2} \end{array}\right)\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right) &amp;amp;= \lambda\left(\begin{array}{c} v_1&lt;br /&gt;
              \\ v_2 \\ v_3 \end{array}\right)&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.36}}&lt;br /&gt;
for each &amp;lt;math&amp;gt;\lambda\,\!&amp;lt;/math&amp;gt;.  &lt;br /&gt;
For &amp;lt;math&amp;gt;\lambda = 3\,\!&amp;lt;/math&amp;gt;, we get the following equations:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\frac{5}{2}v_1 + \frac{i}{2}v_3 &amp;amp;= 3v_1, \\&lt;br /&gt;
2v_2 &amp;amp;= 3v_2,  \\&lt;br /&gt;
-\frac{i}{2}v_1 + \frac{5}{2}v_3 &amp;amp;= 3v_3, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.37}}&lt;br /&gt;
so &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
iv_3  &amp;amp;= v_1, \\&lt;br /&gt;
v_2 &amp;amp;= 0. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.38}}&lt;br /&gt;
Now for &amp;lt;math&amp;gt;\lambda = 2\,\!&amp;lt;/math&amp;gt;, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\frac{5}{2}v_1 +\frac{i}{2}v_3 &amp;amp;= 2 v_1, \\&lt;br /&gt;
2v_2 &amp;amp;= 2v_2, \\&lt;br /&gt;
-\frac{i}{2}v_1 + \frac{5}{2}v_3 &amp;amp;= 2v_3, &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.39}}&lt;br /&gt;
so &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
v_3  &amp;amp;= iv_1, \\&lt;br /&gt;
v_2 &amp;amp;= \mbox{anything}. &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.40}}&lt;br /&gt;
We would like to have a set of orthonormal vectors.  (We can always choose the set to be orthonormal.)  We choose the three eigenvectors to be&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_2 = \left(\begin{array}{c} 1 \\ a \\ i&lt;br /&gt;
  \end{array}\right), \;\;  &lt;br /&gt;
v_2^\prime = \left(\begin{array}{c} 1 \\ a^\prime \\ i&lt;br /&gt;
  \end{array}\right), \;\;&lt;br /&gt;
v_3 = \left(\begin{array}{c} i \\ 0 \\ 1&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
We set the inner product of the two vectors &amp;lt;math&amp;gt; v_2\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; v_2^\prime \,\!&amp;lt;/math&amp;gt; equal to zero so as to have then be orthogonal: &lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
1 + a a^\prime +1 = 2 + a a^\prime = 0.&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Now we can choose &amp;lt;math&amp;gt; a = \sqrt{2}\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt; a^\prime = -\sqrt{2}\,\!&amp;lt;/math&amp;gt; so that the normalized eigenvectors are&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
v_2 = \frac{1}{2}\left(\begin{array}{c} 1 \\ \sqrt{2} \\ i&lt;br /&gt;
  \end{array}\right), \;\;  &lt;br /&gt;
v_2^\prime = \frac{1}{2}\left(\begin{array}{c} 1 \\ -\sqrt{2} \\ i&lt;br /&gt;
  \end{array}\right), \;\;&lt;br /&gt;
v_3 = \frac{1}{\sqrt{2}}\left(\begin{array}{c} i \\ 0 \\ 1&lt;br /&gt;
  \end{array}\right).&lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Tensor Products===&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The tensor product, &amp;lt;!--\index{tensor product} --&amp;gt;&lt;br /&gt;
or the Kronecker product, &amp;lt;!--\index{Kronecker product}--&amp;gt;&lt;br /&gt;
is used extensively in quantum mechanics and&lt;br /&gt;
throughout the course.  It is commonly denoted with a &amp;lt;math&amp;gt;\otimes\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
symbol, although this is often left out.  In fact, the following&lt;br /&gt;
are commonly found in the literature as notation for the tensor&lt;br /&gt;
product of two vectors &amp;lt;math&amp;gt;\left\vert\Psi\right\rangle\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\left\vert\Phi\right\rangle\,\!&amp;lt;/math&amp;gt;:&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert\Psi\right\rangle\otimes\left\vert\Phi\right\rangle &amp;amp;= \left\vert\Psi\right\rangle\left\vert\Phi\right\rangle  \\&lt;br /&gt;
                         &amp;amp;= \left\vert\Psi\Phi\right\rangle.&lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.41}}&lt;br /&gt;
Each of these has its advantages and will all be used in&lt;br /&gt;
different circumstances in this text.  &lt;br /&gt;
&lt;br /&gt;
The tensor product is also often used for operators.  Several&lt;br /&gt;
examples &lt;br /&gt;
will be given, one that explicitly calculates the tensor product for&lt;br /&gt;
two vectors and one that calculates it for two matrices which could&lt;br /&gt;
represent operators.  However, these are not different in the sense&lt;br /&gt;
that a vector is a &amp;lt;math&amp;gt;1\times n\,\!&amp;lt;/math&amp;gt; or an &amp;lt;math&amp;gt;n\times 1\,\!&amp;lt;/math&amp;gt; matrix.  It is also&lt;br /&gt;
noteworthy that the two objects in the tensor product need not be of&lt;br /&gt;
the same type.  In general, a tensor product of an &amp;lt;math&amp;gt;n\times m\,\!&amp;lt;/math&amp;gt; object&lt;br /&gt;
(array) with a &amp;lt;math&amp;gt;p\times q\,\!&amp;lt;/math&amp;gt; object will produce an &amp;lt;math&amp;gt;np\times mq\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
object.  &lt;br /&gt;
 &lt;br /&gt;
&lt;br /&gt;
The tensor product of two objects is computed as follows.&lt;br /&gt;
Let &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt; be an &amp;lt;math&amp;gt;n\times m\,\!&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt; be a &amp;lt;math&amp;gt;p\times q\,\!&amp;lt;/math&amp;gt; array, &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
A = \left(\begin{array}{cccc} &lt;br /&gt;
           a_{11} &amp;amp; a_{12} &amp;amp; \cdots &amp;amp; a_{1m} \\&lt;br /&gt;
           a_{21} &amp;amp; a_{22} &amp;amp; \cdots &amp;amp; a_{2m} \\&lt;br /&gt;
           \vdots &amp;amp;        &amp;amp; \ddots &amp;amp;      \\&lt;br /&gt;
           a_{n1} &amp;amp; a_{n2} &amp;amp; \cdots &amp;amp; a_{nm} \end{array}\right),&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.42}}&lt;br /&gt;
and similarly for &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt;.  Then &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;&lt;br /&gt;
A\otimes B = \left(\begin{array}{cccc} &lt;br /&gt;
             a_{11}B &amp;amp; a_{12}B &amp;amp; \cdots &amp;amp; a_{1m}B \\&lt;br /&gt;
             a_{21}B &amp;amp; a_{22}B &amp;amp; \cdots &amp;amp; a_{2m}B \\&lt;br /&gt;
             \vdots  &amp;amp;         &amp;amp; \ddots &amp;amp;      \\&lt;br /&gt;
             a_{n1}B &amp;amp; a_{n2}B &amp;amp; \cdots &amp;amp; a_{nm}B \end{array}\right).  &lt;br /&gt;
&amp;lt;/math&amp;gt;|C.43}}&lt;br /&gt;
&lt;br /&gt;
Let us now consider two examples.  First let &amp;lt;math&amp;gt;\left\vert\phi\right\rangle\,\!&amp;lt;/math&amp;gt; and&lt;br /&gt;
&amp;lt;math&amp;gt;\left\vert\psi\right\rangle\,\!&amp;lt;/math&amp;gt; be as before,&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
\left\vert\psi\right\rangle = \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right) \;\; &lt;br /&gt;
\mbox{and} \;\; &lt;br /&gt;
\left\vert\phi\right\rangle = \left(\begin{array}{c} \gamma \\ \delta&lt;br /&gt;
  \end{array}\right). &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Then &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert\psi\right\rangle\otimes\left\vert\phi\right\rangle &amp;amp;= \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right) &lt;br /&gt;
\otimes &lt;br /&gt;
\left(\begin{array}{c} \gamma \\ \delta&lt;br /&gt;
  \end{array}\right)   \\&lt;br /&gt;
                            &amp;amp;= \left(\begin{array}{c} \alpha\gamma\\ &lt;br /&gt;
                                                     \alpha\delta \\&lt;br /&gt;
                                                     \beta\gamma \\ &lt;br /&gt;
                                                     \beta\delta &lt;br /&gt;
                                       \end{array}\right).  &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.44}}&lt;br /&gt;
Also&lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
\left\vert\psi\right\rangle\otimes\left\langle\phi\right\vert &amp;amp;= \left\vert\psi\right\rangle\left\langle\phi\right\vert \\&lt;br /&gt;
                            &amp;amp;= \left(\begin{array}{c} \alpha \\ \beta&lt;br /&gt;
  \end{array}\right) &lt;br /&gt;
\otimes &lt;br /&gt;
\left(\begin{array}{cc} \gamma^* &amp;amp; \delta^*&lt;br /&gt;
  \end{array}\right)   \\&lt;br /&gt;
                            &amp;amp;= \left(\begin{array}{cc}&lt;br /&gt;
                                \alpha\gamma^* &amp;amp; \alpha\delta^* \\&lt;br /&gt;
                                \beta\gamma^*  &amp;amp; \beta\delta^* &lt;br /&gt;
                                       \end{array}\right).  &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.45}}&lt;br /&gt;
&lt;br /&gt;
Now consider two matrices&lt;br /&gt;
&amp;lt;center&amp;gt;&amp;lt;math&amp;gt;&lt;br /&gt;
A = \left(\begin{array}{cc} &lt;br /&gt;
                 a &amp;amp; b \\&lt;br /&gt;
                 c &amp;amp; d  \end{array}\right) &lt;br /&gt;
 \;\; \mbox{and} \;\;&lt;br /&gt;
B = \left(\begin{array}{cc} &lt;br /&gt;
               e &amp;amp; f \\&lt;br /&gt;
               g &amp;amp; h  \end{array}\right).  &lt;br /&gt;
\,\!&amp;lt;/math&amp;gt;&amp;lt;/center&amp;gt;&lt;br /&gt;
Then &lt;br /&gt;
{{Equation|&amp;lt;math&amp;gt;\begin{align}&lt;br /&gt;
A\otimes B &amp;amp;=  \left(\begin{array}{cc} &lt;br /&gt;
                 a &amp;amp; b \\&lt;br /&gt;
                 c &amp;amp; d  \end{array}\right) &lt;br /&gt;
 \otimes&lt;br /&gt;
               \left(\begin{array}{cc} &lt;br /&gt;
                 e &amp;amp; f \\&lt;br /&gt;
                 g &amp;amp; h  \end{array}\right)   \\  &lt;br /&gt;
           &amp;amp;=  \left(\begin{array}{cccc} &lt;br /&gt;
                 ae &amp;amp; af &amp;amp; be &amp;amp; bf \\&lt;br /&gt;
                 ag &amp;amp; ah &amp;amp; bg &amp;amp; bh \\&lt;br /&gt;
                 ce &amp;amp; cf &amp;amp; de &amp;amp; df \\&lt;br /&gt;
                 cg &amp;amp; ch &amp;amp; dg &amp;amp; dh \end{array}\right).  &lt;br /&gt;
\end{align}&lt;br /&gt;
&amp;lt;/math&amp;gt;|C.46}}&lt;br /&gt;
&lt;br /&gt;
====Properties of Tensor Products====&lt;br /&gt;
&lt;br /&gt;
Listed here are properties of tensor products that are useful, with &amp;lt;math&amp;gt;A\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;B\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;C\,\!&amp;lt;/math&amp;gt;, &amp;lt;math&amp;gt;D\,\!&amp;lt;/math&amp;gt; of any type:&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)(C\otimes D) = AC \otimes BD\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)^T = A^T\otimes B^T\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)^* = A^*\otimes B^*\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A\otimes B)\otimes C = A\otimes(B\otimes C)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;(A+B) \otimes C = A\otimes C+B\otimes C\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;A\otimes(B+C) = A\otimes B + A\otimes C\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
#&amp;lt;math&amp;gt;\mbox{Tr}(A\otimes B) = \mbox{Tr}(A)\mbox{Tr}(B)\,\!&amp;lt;/math&amp;gt;&lt;br /&gt;
(See [[Bibliography#HornNJohnsonII|Horn and Johnson, Topics in Matrix Analysis [10]]], Chapter 4.)&lt;/div&gt;</summary>
		<author><name>Hilary</name></author>
		
	</entry>
</feed>