Lindbladian

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.