Reduced dynamics explained

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

\rho _{SE} ( 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\rho _{S} (t) = Tr _{E} [U _{t} \rho _{SE} ( 0 ) U ^{\dagger]} _{t} If we assume that the mapping \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\rho _{S} = \sum _{i} F _{i} \rho _{S} ( 0 ) F ^{\dagger} _{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 \rho _{SE} ( 0 ) = \rho _{S} ( 0 ) ⊗ \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]

Notes

1. Pechukas . Philip . Reduced Dynamics Need Not Be Completely Positive . Physical Review Letters . American Physical Society (APS) . 73 . 8 . 1994-08-22 . 0031-9007 . 10.1103/physrevlett.73.1060 . 1060–1062. 10057614 . 1994PhRvL..73.1060P .

References

• Nielsen, Michael A. and Isaac L. Chuang (2000). Quantum Computation and Quantum Information, Cambridge University Press,