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 ρ S E ( 0 ) (which in general may be entangled) and undergoing unitary evolution given by U t . Then the reduced dynamics of the system alone is simply
ρ S ( t ) = T r E [ U t ρ S E ( 0 ) U t † ] If we assume that the mapping ρ S ( 0 ) ↦ ρ 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
ρ S = ∑ i F i ρ S ( 0 ) F i † where the 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 ρ S E ( 0 ) = ρ S ( 0 ) ⊗ ρ E ( 0 ) , it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.
