Theorem: Physical finite evolution

For fixed coefficients, $e^{t\mathcal{L}}$ is a unique CPTP semigroup. The same physical conclusion holds for a finite piecewise-continuous time-dependent coefficient family with nonnegative rates and self-adjoint Hamiltonians.

Proof. The jump operators are $\sqrt{k_{ij}}|i⟩⟨j|$ and $\sqrt{{\gamma }_i}{P}_i$. Equation (eq:qgenerator) has finite GKSL form, so the standard finite-dimensional generator theorem applies GKS,Lindblad. For time-dependent coefficients, approximate on compact intervals by piecewise-constant physical generators. Their channel products converge to the unique finite linear-ODE propagator; complete positivity and trace preservation are closed under this norm limit.