Quantum Dynamical Semigroup

A Quantum Dynamical Semigroup is a set of Dynamical Maps $\{T_t{\}}_{t\ge 0}$ , operating on Density Operators $\rho$, satisfying:

1. ${\rho }_t=T_t{\rho }_0$
2. $T_{t+s}=T_t{T}_s\forall t,s\ge 0$. This is known as the semigroup property.
3. $T_0=\mathbb{I}$, i.e. $T_0$ is the identity map.
4. $T_{t+h}[\rho ]\stackrel{h\to 0}{\to }{T}_t[\rho ]\forall t\ge 0,\forall \rho \in B(H)$, where $B(H)$ is the set of Bounded Linear Operators acting on the Hilbert Space $H$.