Quantum dynamical semigroup

In quantum mechanics, and especially the theory of open quantum systems, a quantum dynamical semigroup is family of quantum dynamical maps $\phi_t$ on the space of density matrices indexed by a single time parameter $t \ge 0$ that obey the semigroup property

$\phi_s(\phi_t(\rho)) = \phi_{t+s}(\rho) , \qquad t,s \ge 0.$

A quantum dynamical semigroup is generated by a Lindblad superoperator. The generator can be obtained by

$\mathcal{L}(\rho) = \mathrm{lim}_{\Delta t \to 0} \frac{\phi_{\Delta t}(\rho)-\phi_0(\rho)}{\Delta t}$

which, by the linearity of $\phi_t$, is a linear superoperator. The semigroup can be recovered as

$\phi_{t+s}(\rho) = e^{\mathcal{L}s} \phi_t(\rho).$