 Lindblad equation

The Lindblad equation describes the generator of a time-homogeneous, completely positive, trace-preserving quantum dynamical semigroup in a finite-dimensional system. In units $\hslash =1$, the Hamiltonian $H$ supplies coherent evolution and the Lindblad operators $L_{\alpha }$ specify dissipative contributions.

A Markovian continuous-time quantum channel obeys

The Lindblad dissipator combines its gain term and two anticommutator terms so that the trace remains one. For $L=\sqrt{\gamma /2}Z$ and $H=0$, diagonal entries in the $Z$ basis stay fixed while an off-diagonal element obeys ${\dot{\rho }}_{01}=-\gamma {\rho }_{01}$: dephasing without population decay. The semigroup assumption is a specific model of environmental memory loss. ^[ref]