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 ) {\displaystyle \rho _{SE}(0)\,} (which in general may be entangled) and undergoing unitary evolution given by U t {\displaystyle 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 † ] {\displaystyle \rho _{S}(t)=\mathrm {Tr} _{E}[U_{t}\rho _{SE}(0)U_{t}^{\dagger }]}

If we assume that the mapping ρ S ( 0 ) ↦ ρ S ( t ) {\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

ρ S = ∑ i F i ρ S ( 0 ) F i † {\displaystyle \rho _{S}=\sum _{i}F_{i}\rho _{S}(0)F_{i}^{\dagger }}

where the F i {\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 ρ S E ( 0 ) = ρ S ( 0 ) ⊗ ρ E ( 0 ) {\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.

Notes

  • Nielsen, Michael A. and Isaac L. Chuang (2000). Quantum Computation and Quantum Information, Cambridge University Press, ISBN 0-521-63503-9