Lindblad equation

A Markovian continuous-time quantum channel obeys

A Lindblad dissipator is the non-Hamiltonian generator associated with a Lindblad operator $V$. It preserves the trace because cyclicity gives $\mathrm{Tr}(V\rho V^{†})=\mathrm{Tr}(V^{†}V\rho )$, cancelling the two anticommutator terms. It preserves Hermiticity and satisfies $\mathcal{D}[V](I)=[V,V^{†}]$; it need not preserve the identity or the purity of a density operator.

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 $\dot{\rho }=\mathcal{L}(\rho )$.

For any initial density operator, average its time-independent Lindblad equation trajectory as ${\bar{\rho }}_T=T^{-1}{\int }_0^T\rho (t)dt$. These averages stay in the compact convex set of density operators. Along a convergent subsequence, $\mathcal{L}({\bar{\rho }}_T)=(\rho (T)-\rho (0))/T\to 0$, so the limit is stationary. Thus a finite-dimensional physical Affine Bloch equation has at least one physical equilibrium, even though an arbitrary affine real equation can have none.

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. In one jump unravelling, a state evolves between jumps 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.