List of symbols#
Symbol |
Description |
|---|---|
\(\eye\) |
imaginary unit |
\(\pi\) |
mathematical constant pi (\(\approx 3.14159\ldots\)) |
\(\e\) |
Euler’s number (\(\approx 2.71828\ldots\)) |
\(\Reals\) |
set of real numbers |
\(\Integers\) |
set of integers |
\(\Naturals\) |
set of natural numbers |
\(\Complexes\) |
set of complex numbers |
\(\oplus\) |
addition operation in modular arithmetic (\(x \oplus y \equiv x + y \mathrel{\mathrm{mod}} m\) for a modulus \(m \in \IntegersPositive\)) |
\(\ominus\) |
subtraction operation in modular arithmetic (\(x \ominus y \equiv x - y \mathrel{\mathrm{mod}} m\) for a modulus \(m \in \IntegersPositive\)) |
\(\otimes\) |
tensor product operation, Kronecker product operation |
\(\SpaceVector\) |
vector space |
\(\dim({}\cdot{})\) |
dimension |
\(\Span({}\cdot{})\) |
(linear) span |
\(\overline{{}\cdot{}}\) |
complex conjugate (the mathematician’s notation) |
\(\conj{{}\cdot{}}{}\) |
complex conjugate (the physicist’s notation) |
\({{}\cdot{}}^\transpose\) |
transpose |
\({{}\cdot{}}^\dagger\) |
Hermitian conjugate (conjugate transpose) |
\(\norm{{}\cdot{}}_p\) |
vector \(p\)-norm |
\(\ell^p\) |
space of sequences with converging \(p\)-norm |
\(\SpaceLebesgue^p\) |
space of functions with converging \(p\)-norm |
\(\SpaceHilbert\) |
Hilbert space |
\(\SpacePure(\SpaceHilbert)\) |
space of pure quantum states on \(\SpaceHilbert\) |
\(\SpaceMixed(\SpaceHilbert)\) |
space of mixed quantum states on \(\SpaceHilbert\) |
\(\SpaceLinear(\SpaceHilbert)\) |
space of linear operators on \(\SpaceHilbert\) |
\(\psi, \phi, \chi, \ldots\) |
vectors (lowercase Greek letters) |
\(\Basis_i, \Basis_j, \ldots\) |
basis vectors |
\(\delta_{ij}\) |
Kronecker delta |
\([{}\cdot{},{}\cdot{}]\) |
commutator |
\(\{{}\cdot{},{}\cdot{}\}\) |
anticommutator |
\(\langle{}\cdot{},{}\cdot{}\rangle\) |
inner product (the mathematician’s notation) |
\(\braket{{}\cdot{}}{{}\cdot{}}\) |
inner product (the physicist’s notation) (a “bra-ket”) |
\(\ket{{}\cdot{}}\) |
quantum vector state (a “ket”) |
\(\bra{{}\cdot{}}\) |
quantum covector state (a “bra”) |
\(\ket{i}, \ket{j}, \ldots\) |
number state (e.g., \(\ket{0},\ket{1},\ket{2},\ldots\)) |
\(\ket{\Phi}, \ket{\Psi}\) |
maximally entangled state |
\(\op{{}\cdot{}}\) |
(linear) operator |
\(\op{A}, \op{B}, \op{C}, \ldots\) |
linear operators (uppercase Latin letters) |
\(\op{\rho}, \op{\tau}, \op{\omega}, \ldots\) |
density operators (lowercase Greek letters) |
\(\Identity_{\Dimension}\) |
\(\Dimension \times \Dimension\) identity operator, (unnormalized) maximally mixed state |
\(\op{N}\) |
normal operator |
\(\op{H}\) |
Hermitian operator |
\(\op{S}\) |
skew-Hermitian operator |
\(\op{U}\) |
unitary operator |
\(\op{P}\) |
positive operator |
\(\op{\Pi}\) |
projection operator |
\(\op{X}, \op{Y}, \op{Z}\) |
Pauli gates |
\(\Rotation(\theta)\) |
rotation gate (with angle parameter \(\theta\)) |
\(\Phase(\omega)\) |
phase gate (with phase parameter \(\omega\)) |
\(\op{Z}\) |
\(\pi\)-phase gate (equivalent to Pauli-\(Z\) gate) |
\(\op{S}\) |
\(\tfrac{\pi}{2}\)-phase gate |
\(\op{T}\) |
\(\tfrac{\pi}{4}\)-phase gate |
\(\Swap(p)\) |
SWAP gate (with power parameter \(p\)) |
\(\SUM(n)\) |
SUM gate (with shift parameter \(n\)) |
\(\NOT\) |
NOT gate |
\(\Hadamard\) |
Hadamard gate |
\(\QFT\) |
quantum Fourier transform gate |
\(\VacuumSwap\), \(\VacuumRotation\), \(\VacuumIdentity\) |
vacuum-adapted (\(\ket{0}\)-inclusive) gate variants |
\(\Control^\alpha \Anticontrol^\beta \Unitary^\gamma\) |
controlled-anticontrolled-unitary gate (indices: \(\alpha=\) control, \(\beta=\) anticontrol, \(\gamma=\) unitary) |
\(\Pauli_\mu\) |
Pauli matrices (with identity \(\Pauli_0 = \Identity\)) |
\(\GellMann_\mu\) |
Gell-Mann matrices (with identity \(\GellMann_0 = \Identity\)) |
\(\Kraus\) |
Kraus operator |
\(\SetKraus\) |
set of Kraus operators |
\(p_i\) |
probability (density) of a discrete state (labelled with \(i\)) |
\(\SetProbability\) |
probability distribution (set of probabilities of discrete states) |
\(\SetUnitary\) |
set of unitary operators |
\(\SetObservable\) |
POVM (positive operator-valued measure) (set of positive operators) |
\(\left\langle{}\cdot{}\right\rangle\) |
expected value of an observable |
\(\trace[{}\cdot{}]\) |
trace |
\(\trace_{\{i\}}[{}\cdot{}]\) |
partial trace (over subsystems with labels \(\{i\}_i\)) |
\(\Purity({}\cdot{})\) |
purity |
\(\TraceDistance({}\cdot{} , {}\cdot{})\) |
trace distance |
\(\Fidelity({}\cdot{} , {}\cdot{})\) |
fidelity |
\(\Entropy({}\cdot{})\) |
von Neumann entropy |
\(\Entropy({}\cdot{} \! \mathrel{\Vert} \! {}\cdot{})\) |
relative entropy |
\(\MutualInformation({}\cdot{} : {}\cdot{})\) |
mutual information |
\(\Decoherence({}\cdot{})\) |
decoherence channel |
\(\Depolarization({}\cdot{})\) |
depolarization channel |
\(\MapGeneral_{\Unitary}[{}\cdot{}]\) |
general quantum evolution map |
\(\MapDCTCsCR_{\Unitary}[{}\cdot{},{}\cdot{}]\) |
D-CTCs CR map |
\(\MapDCTCsCV_{\Unitary}[{}\cdot{},{}\cdot{}]\) |
D-CTCs CV map |
\(\MapPCTCsCR_{\Unitary}[{}\cdot{}]\) |
P-CTCs CR map |
\(\MapPCTCsCV_{\Unitary}[{}\cdot{}]\) |
P-CTCs CV map |
\(\OperatorPCTC\) |
P-CTC operator |
\(\Action[\vec{\Position}(\Time)]\) |
classical action over a path \(\vec{\Position}(\Time)\) |
\(\Propagator(\vec{\PositionFinal},\TimeFinal;\vec{\PositionInitial},\TimeInitial)\) |
(non-relativistic) quantum propagator from \((\TimeInitial,\vec{\PositionInitial})\) to \((\TimeFinal,\vec{\PositionFinal})\) |
\(\DifferentialPath\vec{\Position}(\Time)\) |
path-integral spatial differential |
\(\Time\) |
time coordinate |
\(\vec{\Position}\) |
position vector |
\(\vec{\Momentum}\) |
momentum vector |
\(\OperatorHamiltonian\) |
Hamiltonian operator |
\(\OperatorLagrangian\) |
Lagrangian operator |
\(\OperatorKinetic\) |
kinetic energy operator |
\(\OperatorPotential\) |
potential energy operator |
\(\Probability({}\cdot{})\) |
probability density of a continuous function |
\(\Expected({}\cdot{})\) |
expected value of a continuous function |
\(\Mouth^\pm\) |
wormhole mouths (\(\MouthFuture=\) future, \(\MouthPast=\) past) |