Notation

Symbol				Definition
${\mathcal {H}}\,\!$				Hilbert Space
${\mathbb {C} }\,\!$				The set of complex numbers
${\mathbb {R} }\,\!$				The set of real numbers
		Chapter 2 - Qubits and Collections of Qubits
$H\,\!$				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)
$\hbar \,\!$				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)
$L\,\!$				Local operations (see Eq. 4.11)
$L\left\vert \Phi \right\rangle$				Local transformations (see Eq. 4.12)
$\left\vert \psi _{+}\right\rangle ,\left\vert \psi _{-}\right\rangle ,$
$\left\vert \phi _{+}\right\rangle ,\left\vert \phi _{-}\right\rangle$				Bell States (see Eq. 4.14)
${\mbox{Tr}}_{B}(\rho _{ss})\,\!$				The partial trace over one of the subsystems (particle states) of a composite system (see Eq. 4.19)
		Chapter 5 - Quantum Information
$\sigma _{i}^{2}\,\!$				The variance of an observable (see Eq. 5.4)
$H\,\!$				The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)
$\left\vert H\right\rangle \,\!$, $\left\vert V\right\rangle \,\!$
$\left\vert D\right\rangle \,\!$, $\left\vert -D\right\rangle \,\!$				Polarization states of photons (see Figure 5.1 and Table 5.1)
		Chapter 6 - Noise in Quantum Systems
$\rho ^{\prime }\,\!$, $\rho \,\!$				Linear mapping vectors (see Eqs. 6.1 - 6.14)
$A\,\!$				The mapping matrix (see Eqs. 6.6 - 6.12)
$H_{S}\,\!$				The Hamiltonian for the system alone (see Eq. 6.15)
$H_{B}\,\!$				The Hamiltonian for the bath alone (see Eq. 6.15)
$S_{\gamma }\,\!$				Operator on the system (see Eq. 6.15)
$B_{\gamma }\,\!$				Operator on the bath (see Eq. 6.15)
$\rho _{S}(0)\,\!$				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
$A_{\alpha }\,\!$				Operator element
$P\,\!$				A projector onto the code space
$C(n,t)\;\!$				The binomial coefficient (see Eq. 7.17)
$m\;\!$				The number of code words (see Eq. 7.17)
${\mathcal {S}}\,\!$				The stabilizer
${\mathcal {S}}\subset {\mathcal {P}}_{n}\,\!$				An abelian subgroup of the Pauli group that does not contain $-\mathbb {I} ,\pm i\mathbb {I} \,\!$
${\mathcal {C}}({\mathcal {S}})\,\!$				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
$X_{n}\,\!$, $Y_{n}\,\!$, $Z_{n}\,\!$				The Pauli x-operation, y-operation, and z-operation on the nth qubit (see Eq. 8.27)
${\bar {X}}\,\!$				A logical $X\,\!$ operation (see Eqs. 8.28 - 8.34)
${\bar {Z}}\,\!$				A logical $Z\,\!$ operation (see Eqs. 8.29 - 8.35)
${\bar {Y}}\,\!$				A logical $Y\,\!$ operation (see Section 8.5.2)
${\bar {ZZ}}\,\!$				A logical two-qubit entangling gate (see Eq. 8.30)
$H_{ex}^{i,j}\,\!$				The Heisenberg exchange interaction Hamiltonian between two qubits labelled $i\,\!$ and $j\,\!$ (see Eq. 8.31)
$E_{ij}\,\!$				An operator resulting from the exponential of the exchange operation between qubits $i\,\!$ and $j\,\!$ for a time $\pi /4\,\!$ (see Section 8.5.2 and Eq. 8.32 - 8.35)
		Chapter 9 - Dynamical Decoupling Controls
$H(t)\,\!$				A Hermitian matrix (see Eqs. 9.2 - 9.8)
$U(t)\,\!$				A unitary matrix (see Eqs. 9.3 - 9.6)
${\mathcal {T}}\,\!$				The time-ordered exponential (see Eqs. 9.5)
$T\,\!$				Some characteristic time scale (see Eqs. 9.8)
$N\,\!$				Number of different controls to be used
$U_{n}\,\!$				A given control
$H\,\!$				Free evolution given by Eq. 9.1
$H_{eff}\,\!$				The effective Hamiltonian (see Eq. 9.12 and Eqs. 9.14 - 9.18)
$H_{i}\,\!$				The Hamiltonian for the free evolution (see Eq. 9.13)
$B\,\!$				The bath operator (see Eq. 9.13)
$\sigma _{z}\,\!$				Indicates a phase error (see Eq. 9.13)
$U_{X}(\pi )\,\!$				A decoupling pulse, denoted $U_{1}=U_{X}(\pi )\,\!$ (see Eq. 9.14)
$U_{0}\,\!$				The identity, denoted $U_{0}=I\,\!$ (see Eq. 9.14)
$U_{k}\,\!$				Some particular element of the group (see Eqs. 9.19 - 9.22)
$C\,\!$				Some constant (as of yet unknown) (see Eq. 9.19 - 9.22)
$\{\lambda _{i}\}\,\!$				A complete set of Hermitian matrices (see Eqs. 9.23 - 9.25, Section C.3.8, and Section C.6.1)
$a_{i}\,\!$				Set of coefficients (see Eqs. 9.23 - 9.26)
		Chapter 10 - Fault-Tolerant Quantum Computing