Reduced dynamics

In quantum mechanics, especially in the study of open quantum systems, reduced dynamics refers to the time evolution of a density matrix for a system coupled to an environment. Consider a system and environment initially in the state ${\displaystyle \rho _{SE}(0)\,}$ (which in general may be entangled) and undergoing unitary evolution given by ${\displaystyle U_{t}\,}$. Then the reduced dynamics of the system alone is simply

${\displaystyle \rho _{S}(t)=\mathrm {Tr} _{E}[U_{t}\rho _{SE}(0)U_{t}^{\dagger }]}$

If we assume that the mapping ${\displaystyle \rho _{S}(0)\mapsto \rho _{S}(t)}$ is linear and completely positive, then the reduced dynamics can be represented by a quantum operation. This mean we can express it in the operator-sum form

${\displaystyle \rho _{S}=\sum _{i}F_{i}\rho _{S}(0)F_{i}^{\dagger }}$

where the ${\displaystyle F_{i}\,}$ are operators on the Hilbert space of the system alone, and no reference is made to the environment. In particular, if the system and environment are initially in a product state ${\displaystyle \rho _{SE}(0)=\rho _{S}(0)\otimes \rho _{E}(0)}$, it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.¹