Quantum dynamical semigroup

Remarks

Example: on $\C^2$ with $\gamma > 0$, let $T_t(\rho) = \tfrac{1 + e^{-\gamma t}}{2}\rho + \tfrac{1 - e^{-\gamma t}}{2} Z\rho Z$, where $Z = \operatorname{diag}(1, -1)$. Each $T_t$ is a mixture of the unitary conjugations by $I$ and $Z$, so it is a channel. It keeps the diagonal of $\rho$ and multiplies the off-diagonal entries by $e^{-\gamma t}$. Since $e^{-\gamma s}e^{-\gamma t} = e^{-\gamma(s+t)}$ and $T_t(\rho) \to \rho$ as $t \to 0^+$, clauses 2 and 3 hold. Non-example: the channels $S_t(\rho) = \tfrac{1 + \cos t}{2}\rho + \tfrac{1 - \cos t}{2}Z\rho Z$ satisfy $S_0 = \id$ and multiply the off-diagonal entries by $\cos t$. Then $S_{\pi/2} \circ S_{\pi/2}$ multiplies them by $0$ while $S_\pi$ multiplies them by $-1$, so clause 2 fails. When $\mathcal{H}$ is finite-dimensional, $T_t = e^{t\mathcal{L}}$ for a generator $\mathcal{L}$, and the Gorini–Kossakowski–Sudarshan–Lindblad theorem gives the possible forms of $\mathcal{L}$.