Notation

Symbol				Definition
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal H}\,\!}				Hilbert Space
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb{C}}\,\!}				The set of complex numbers
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb{R}}\,\!}				The set of real numbers
		Chapter 2 - Qubits and Collections of Qubits
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\,\!}				The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)
$X\,\!$				The Pauli X matrix (see Table 2.1)
$Y\,\!$				The Pauli Y matrix (see Table 2.1)
$Z\,\!$				The Pauli Z matrix (see Table 2.1)
${\mathbb {I} }\,\!$				Identity operator
${\delta _{ij}}\,\!$				The Kronecker delta (see Equation C.17)
${\epsilon _{ijk}}\,\!$				Epsilon (see Equation C.8)
$\otimes \,\!$				The Tensor Product (see Section C.7)
		Chapter 3 - Physics of Quantum Information
$H\,\!$				The Hamiltonian (see Eqs. 3.1 - 3.4)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\,\!}				H-bar (Planck's constant divided by $2\pi \,\!$)
$U\,\!$				The Unitary Matrix (see Eqs. 3.5 - 3.6)
$\rho \,\!$				The Density Matrix or Density Operator (see Appendix E - Density Operator: Extensions)
${\mbox{Tr}}\,\!$				The trace of a matrix (see Section C.3.5)
$\det(N)\,\!$				The determinant of a matrix (see Section C.3.6)
${\vec {n}}\,\!$				A unit vector (see Section C.5.1)
$\lambda _{\pm }$				Eigenvalues (see Section C.6)
$\langle {\mathcal {O}}\rangle$				The expectation value of an operator (see Eqs. 3.47 - 3.48)
		Chapter 4 - Entanglement
$\lambda \,\!$				A possible hidden variable (see Eqs. 4.4 - 4.9)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\,\!}				Local operations (see Eq. 4.11)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\left\vert \Phi\right\rangle}				Local transformations (see Eq. 4.12)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert \psi_+\right\rangle, \left\vert \psi_-\right\rangle,}
$\left\vert \phi _{+}\right\rangle ,\left\vert \phi _{-}\right\rangle$				Bell States (see Eq. 4.14)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tr}_B(\rho_{ss})\,\!}				The partial trace over one of the subsystems (particle states) of a composite system (see Eq. 4.19)
		Chapter 5 - Quantum Information
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2_i\,\!}				The variance of an observable (see Eq. 5.4)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\,\!}				The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert H\right\rangle\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert V\right\rangle\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert D\right\rangle\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert -D\right\rangle\,\!}				Polarization states of photons (see Figure 5.1 and Table 5.1)
		Chapter 6 - Noise in Quantum Systems
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^\prime\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho\,\!}				Linear mapping vectors (see Eqs. 6.1 - 6.14)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\,\!}				The mapping matrix (see Eqs. 6.6 - 6.12)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{S}\,\!}				The Hamiltonian for the system alone (see Eq. 6.15)
$H_{B}\,\!$				The Hamiltonian for the bath alone (see Eq. 6.15)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\gamma }\,\!}				Operator on the system (see Eq. 6.15)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{\gamma }\,\!}				Operator on the bath (see Eq. 6.15)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_S (0)\,\!}				The initial density matrix of the (open) system (see Eq. 6.16)
$\rho _{B}(0)\,\!$				The initial density matrix of the bath (see Eq. 6.16)
		Chapter 7 - Quantum Error Correcting Codes
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_\alpha\,\!}				Operator element
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P\,\!}				A projector onto the code space
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(n,t)\;\!}				The binomial coefficient (see Eq. 7.17)
$m\;\!$				The number of code words (see Eq. 7.17)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}\,\!}				The stabilizer
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{S}\subset \mathcal{P}_n \,\!}				An abelian subgroup of the Pauli group that does not contain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\mathbb{I},\pm i\mathbb{I}\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{C}(\mathcal{S})\,\!}				A stabilizer code
$P_{1}\,\!$				Classical parity check matrix (see Eq. 7.21)
$G\,\!$				The generator matrix (see Eq. 7.22)
		Chapter 8 - Decoherence-Free/Noiseless Subsystems
$H_{S}\,\!$				Hamiltonian acting only on the system (see Eq. 8.1)
$H_{B}\,\!$				Hamiltonian acting only on the bath (see Eq. 8.1)
$H_{I}\,\!$				The interaction Hamilton (see Eqs. 8.1 and 8.2)
${\mathcal {A}}\,\!$				Denotes the "error algebra" generated by the set $\{H_{S},S_{\alpha }\}\,\!$
$S_{\alpha }\,\!$				Set of error operators acting only on the system (see Eq. 8.2)
$B_{\alpha }\,\!$				Acts only on the bath (see Eq. 8.2)
$U_{dns}\,\!$				A particular unitary transformation
$H_{cpe}\,\!$				Collective phase error Hamiltonian (see Eq. 8.3)
$Z^{(i)}\,\!$				A phase operator which acts on the $i\,\!$th qubit (see Eq. 8.3)
$S_{x}\,\!$, $S_{y}\,\!$, $S_{z}\,\!$				The three collective errors (see Eq. 8.9)
$U_{dns}\,\!$				Unitary transformation corresponding to the collection of Wigner-Clebsch-Gordan coefficients
${\mathcal {C}}^{\perp }\,\!$				The orthogonal subspace (to the code) (see Eq. 8.17)
$\Psi \,\!$				An encoded state
$\{\Lambda _{i}\}\,\!$				Basis for the noise operators
$S\in {\mathcal {S}}\,\!$				The stabilizer element (see Eq. 8.18)
$C\,\!$				Casimir operator (see Eqs. 8.22 - 8.24)
$\lambda _{\alpha }\,\!$				Elements of the Lie algebra (see Eqs. 8.22 - 8.26, Eq. D.21, Section D.7.1 and Sections D.8.1 - D.8.3)
$H\,\!$				The Hamiltonian
$\mu _{ijk}\,\!$				A complete set of Hermitian matrices in terms of which any Hermitian matrix can be expanded
$S_{i}\,\!$				Form a basis for the stabilizer of the system
$\sum _{i}g_{i}S_{i}\,\!$				An arbitrary linear combination of those stabilizer elements
$a_{ijk}\,\!$				A set of real numbers
$g_{i}\,\!$				Arbitrary coefficients
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_n \,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_n \,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_n \,\!}				The Pauli x-operation, y-operation, and z-operation on the nth qubit (see Eq. 8.27)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{X} \,\!}				A logical Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \,\!} operation (see Eqs. 8.28 - 8.34)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Z} \,\!}				A logical Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z \,\!} operation (see Eqs. 8.29 - 8.35)
${\bar {Y}}\,\!$				A logical $Y\,\!$ operation (see Section 8.5.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{ZZ} \,\!}				A logical two-qubit entangling gate (see Eq. 8.30)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{ex}^{i,j} \,\!}				The Heisenberg exchange interaction Hamiltonian between two qubits labelled Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \,\!} and $j\,\!$ (see Eq. 8.31)
$E_{ij}\,\!$				An operator resulting from the exponential of the exchange operation between qubits Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j \,\!} for a time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/4\,\!}