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)

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A unit vector (see Section C.5.1)

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Eigenvalues (see Section C.6)

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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)

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Local operations (see Eq. 4.11)

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Local transformations (see Eq. 4.12)

$\left\vert \psi _{+}\right\rangle ,\left\vert \psi _{-}\right\rangle ,$

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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

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The variance of an observable (see Eq. 5.4)

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The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)

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Polarization states of photons (see Figure 5.1 and Table 5.1)

Chapter 6 - Noise in Quantum Systems

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Linear mapping vectors (see Eqs. 6.1 - 6.14)

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The mapping matrix (see Eqs. 6.6 - 6.12)

$H_{S}\,\!$

The Hamiltonian for the system alone (see Eq. 6.15)

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The Hamiltonian for the bath alone (see Eq. 6.15)

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Operator on the system (see Eq. 6.15)

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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

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Operator element

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A projector onto the code space

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The binomial coefficient (see Eq. 7.17)

$m\;\!$

The number of code words (see Eq. 7.17)

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The stabilizer

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${\mathcal {C}}({\mathcal {S}})\,\!$

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)

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A particular unitary transformation

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Collective phase error Hamiltonian (see Eq. 8.3)

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A phase operator which acts on the $i\,\!$th qubit (see Eq. 8.3)

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The three collective errors (see Eq. 8.9)

$U_{dns}\,\!$

Unitary transformation corresponding to the collection of Wigner-Clebsch-Gordan coefficients

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The orthogonal subspace (to the code) (see Eq. 8.17)

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An encoded state

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Basis for the noise operators

$S\in {\mathcal {S}}\,\!$

The stabilizer element (see Eq. 8.18)

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Casimir operator (see Eqs. 8.22 - 8.24)

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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)

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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

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An arbitrary linear combination of those stabilizer elements

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A set of real numbers

$g_{i}\,\!$

Arbitrary coefficients

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The Pauli x-operation, y-operation, and z-operation on the nth qubit (see Eq. 8.27)

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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)

${\bar {Z}}\,\!$

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 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 \,\!} operation (see Section 8.5.2)

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A logical two-qubit entangling gate (see Eq. 8.30)

$H_{ex}^{i,j}\,\!$

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 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 \,\!} (see Eq. 8.31)

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An operator resulting from the exponential of the exchange operation between qubits $i\,\!$ and $j\,\!$ 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\,\!} (see Section 8.5.2 and Eq. 8.32 - 8.35)

Chapter 9 - Dynamical Decoupling Controls