Notation

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Symbol Definition
Hilbert Space
The set of complex numbers
The set of real numbers
Chapter 2 - Qubits and Collections of Qubits
The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)
The Pauli X matrix (see Table 2.1)
The Pauli Y matrix (see Table 2.1)
The Pauli Z matrix (see Table 2.1)
Identity operator
The Kronecker delta (see Equation C.17)
Epsilon (see Equation C.8)
The Tensor Product (see Section C.7)
Chapter 3 - Physics of Quantum Information
The Hamiltonian (see Eqs. 3.1 - 3.4)
H-bar (Planck's constant divided by )
The Unitary Matrix (see Eqs. 3.5 - 3.6)
The Density Matrix or Density Operator (see Appendix E - Density Operator: Extensions)
The trace of a matrix (see Section C.3.5)
The determinant of a matrix (see Section C.3.6)
A unit vector (see Section C.5.1)
Eigenvalues (see Section C.6)
The expectation value of an operator (see Eqs. 3.47 - 3.48)
Chapter 4 - Entanglement
A possible hidden variable (see Eqs. 4.4 - 4.9)
Local operations (see Eq. 4.11)
Local transformations (see Eq. 4.12)
Bell States (see Eq. 4.14)
The partial trace over one of the subsystems (particle states) of a composite system (see Eq. 4.19)
Chapter 5 - Quantum Information
The variance of an observable (see Eq. 5.4)
The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)
,
, Polarization states of photons (see Figure 5.1 and Table 5.1)
Chapter 6 - Noise in Quantum Systems
, Linear mapping vectors (see Eqs. 6.1 - 6.14)
The mapping matrix (see Eqs. 6.6 - 6.12)
The Hamiltonian for the system alone (see Eq. 6.15)
The Hamiltonian for the bath alone (see Eq. 6.15)
Operator on the system (see Eq. 6.15)
Operator on the bath (see Eq. 6.15)
The initial density matrix of the (open) system (see Eq. 6.16)
The initial density matrix of the bath (see Eq. 6.16)
Chapter 7 - Quantum Error Correcting Codes
Operator element
A projector onto the code space
The binomial coefficient (see Eq. 7.17)
The number of code words (see Eq. 7.17)
The stabilizer
An abelian subgroup of the Pauli group that does not contain
A stabilizer code
Classical parity check matrix (see Eq. 7.21)
The generator matrix (see Eq. 7.22)
Chapter 8 - Decoherence-Free/Noiseless Subsystems
Hamiltonian acting only on the system (see Eq. 8.1)
Hamiltonian acting only on the bath (see Eq. 8.1)
The interaction Hamilton (see Eqs. 8.1 and 8.2)
Denotes the "error algebra" generated by the set
Set of error operators acting only on the system (see Eq. 8.2)
Acts only on the bath (see Eq. 8.2)
A particular unitary transformation
Collective phase error Hamiltonian (see Eq. 8.3)
A phase operator which acts on the th qubit (see Eq. 8.3)
, , The three collective errors (see Eq. 8.9)
Unitary transformation corresponding to the collection of Wigner-Clebsch-Gordan coefficients
The orthogonal subspace (to the code) (see Eq. 8.17)
An encoded state
Basis for the noise operators
The stabilizer element (see Eq. 8.18)
Casimir operator (see Eqs. 8.22 - 8.24)
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)
The Hamiltonian
A complete set of Hermitian matrices in terms of which any Hermitian matrix can be expanded
Form a basis for the stabilizer of the system
An arbitrary linear combination of those stabilizer elements
A set of real numbers
Arbitrary coefficients
, , The Pauli x-operation, y-operation, and z-operation on the nth qubit (see Eq. 8.27)
A logical operation (see Eqs. 8.28 - 8.34)
A logical operation (see Eqs. 8.29 - 8.35)
A logical operation (see Section 8.5.2)
A logical two-qubit entangling gate (see Eq. 8.30)
The Heisenberg exchange interaction Hamiltonian between two qubits labelled and (see Eq. 8.31)
An operator resulting from the exponential of the exchange operation between qubits and for a time (see Section 8.5.2 and Eq. 8.32 - 8.35)
Chapter 9 - Dynamical Decoupling Controls
A Hermitian matrix (see Eqs. 9.2 - 9.8)
A unitary matrix (see Eqs. 9.3 - 9.6)
The time-ordered exponential (see Eqs. 9.5)
Some characteristic time scale (see Eqs. 9.8)
Number of different controls to be used
A given control
Free evolution given by Eq. 9.1
The effective Hamiltonian (see Eq. 9.12 and Eqs. 9.14 - 9.18)
The Hamiltonian for the free evolution (see Eq. 9.13)
The bath operator (see Eq. 9.13)
Indicates a phase error (see Eq. 9.13)
A decoupling pulse, denoted (see Eq. 9.14)
The identity, denoted (see Eq. 9.14)
Some particular element of the group (see Eqs. 9.19 - 9.22)
Some constant (as of yet unknown) (see Eq. 9.19 - 9.22)
A complete set of Hermitian matrices (see Eqs. 9.23 - 9.25, Section C.3.8, and Section C.6.1)
Set of coefficients (see Eqs. 9.23 - 9.26)
Chapter 10 - Fault-Tolerant Quantum Computing
Error probability for one physical qubit
Shor proposed ancilla state (see Eq. 10.1)
Steane proposed ancilla state (see Eq. 10.2)
Chapter 11 - Hybrid Methods of Quantum Error Prevention
An encoded DFS/NS zero state (see Eq. 11.1)
The corresponding DFS/NS encoded one state (see Eq. 11.1)
The system (see Eq. 11.2)
The gauge system (see Eq. 11.2)
An error (see Eq. 11.2)
An error recovery operation (see Eq. 11.2)
Chapter 12 - Conclusions and Further Study
Chapter 13 - Topological Quantum Error Correction
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