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

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

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