Completely positive map (CP map)

A linear map $\Phi$ between matrix algebras is completely positive when $\Phi \otimes {\mathrm{id}}_r$ is positive for every ancillary dimension $r$. In finite dimensions this is equivalent to positivity of its Choi matrix and to the existence of a Kraus representation.

Every finite-dimensional completely positive map has $T(\rho )={\sum }_j{K}_j\rho K_j^{†}$. It is trace preserving exactly when ${\sum }_j{K}_j^{†}{K}_j=I$.

The operators $K_j$ in a Kraus representation are Kraus operators. For a measurement outcome $i$, its associated Kraus operators determine both the outcome probability and the conditional post-measurement state.

For normalized maximally entangled $|\varphi ⟩=d^{-1/2}{\sum }_j|jj⟩$, the Choi matrix of $T$ is $C_T=(T\otimes \mathrm{id})(|\varphi ⟩⟨\varphi |)$. Choi's theorem says $T$ is completely positive exactly when $C_T\ge 0$.

A quantum channel is a linear completely positive trace-preserving map between operator algebras.

A random unitary channel has $T(\rho )={\sum }_j{p}_j{U}_j\rho U_j^{†}$. Its Choi matrix is a convex combination of maximally entangled pure states.

A one-qubit Pauli channel applies $I,{\sigma }_x,{\sigma }_y,{\sigma }_z$ with probabilities $p_0,p_x,p_y,p_z$. It maps a Bloch vector diagonally by independently contracting or reversing its three components.

The phase-flip channel is ${\Lambda }_p(\rho )=(1-p)\rho +p{\sigma }_z\rho {\sigma }_z$. It multiplies the $x$ and $y$ components of the Bloch vector by $1-2p$ and leaves the $z$ component fixed.

A dephasing channel suppresses off-diagonal density-matrix entries in a preferred basis while preserving populations. Complete dephasing is identical to a projective measurement in that basis with its outcome discarded.

A quantum channel is unital when $T(I)=I$. Every random unitary channel is unital, while the converse fails in dimension at least three.

The Werner–Holevo channel is $T(\rho )=(\mathrm{Tr}(\rho )I-{\rho }^T)/(d-1)$. Its normalized Choi matrix is the maximally mixed state on the antisymmetric subspace.

A quantum channel is strictly contractive in trace distance when it reduces the distance between every pair of distinct density operators by a uniform factor smaller than one.

A finite-dimensional quantum channel is primitive when some power maps every nonzero positive operator to a positive-definite operator. Equivalently, eigenvalue one is simple and no other eigenvalue lies on the unit circle.

A Stinespring dilation represents a quantum channel by an isometry $V={\sum }_i{K}_i\otimes |i⟩$ followed by tracing out the environment.

For a purification $|{\Psi }_{\rho }{⟩}_{RA}$, the entanglement fidelity of a channel $\Lambda$ isIt is independent of the chosen purification.

A Markovian continuous-time quantum channel obeys

The Lindblad operators $L_{\alpha }$ describe dissipative channels in a Markovian open-system evolution. Their normalization fixes the corresponding rates, and their representation is not unique.

The Lindbladian $\mathcal{L}$ is the linear superoperator on density operators defined by the right-hand side of a Lindblad equation, so that $\dot{\rho }=\mathcal{L}(\rho )$.

The Lindbladian gap is the smallest positive decay rate $-\mathrm{Re}\lambda$ among nonzero eigenvalues $\lambda$ of a relaxing Lindbladian. It controls the slowest asymptotic exponential approach to the stationary state.

A finite-dimensional Lindbladian converges to one stationary state for every initial state exactly when zero is a simple eigenvalue and every other eigenvalue has strictly negative real part. A trivial commutant of the Hamiltonian and all jump operators and their adjoints is a standard irreducibility criterion leading to uniqueness under the usual finite-dimensional hypotheses.

A quantum-trajectory unravelling represents a Lindblad equation as an ensemble of stochastic pure-state histories. Between jumps a state evolves under the non-Hermitian effective Hamiltonian $H_{\mathrm{eff}}=H-\frac{i}{2}{\sum }_{\alpha }{L}_{\alpha }^{†}{L}_{\alpha }$, while a jump of type $\alpha$ sends it to a normalized multiple of $L_{\alpha }|\psi ⟩$.

A pure stationary state is dark when every jump operator acts on it as a scalar, after an allowed shift of the jump operators, and the corresponding effective Hamiltonian preserves its ray. Every trajectory that reaches such a state remains on that same ray.