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G. Lindblad characterized generators of completely positive quantum dynamical semigroups. In finite-dimensional trace-preserving dynamics, the corresponding Lindblad equation can be written, with $\hslash =1$,where $H$ is a Quantum Hamiltonian, $\rho$ a density matrix and $L_a$ the Lindblad operators.^[ref]

Complete positivity preserves positivity even when the system is considered together with an untouched auxiliary quantum system. The semigroup and continuity assumptions are substantive: the displayed time-independent Markovian form does not describe every possible reduced evolution with memory.

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]

A quantum dynamical semigroup is a continuous family $({\Phi }_t{)}_{t\ge 0}$ of linear maps on density matrices satisfying ${\Phi }_0=I$ and ${\Phi }_{t+s}={\Phi }_t\circ {\Phi }_s$, with each map completely positive and trace preserving. The composition law expresses time-homogeneous evolution with no dependence on how the interval is divided.

In finite dimensions, continuity gives ${\Phi }_t=e^{t\mathcal{L}}$, whose generator has Lindblad equation form. For a qubit dephasing example,Off-diagonal decay factors multiply under composition, while populations stay fixed. The map is the mixture $(1+e^{-\gamma t})\rho /2+(1-e^{-\gamma t})Z\rho Z/2$, which verifies complete positivity. A semigroup need not have physical inverse maps at positive times: undoing this decay would generally amplify coherence beyond valid states.^[ref]