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 [math]\displaystyle{ \rho_{SE} (0) \, }[/math] (which in general may be entangled) and undergoing unitary evolution given by [math]\displaystyle{ U_t \, }[/math]. Then the reduced dynamics of the system alone is simply

[math]\displaystyle{ \rho_S (t) = \mathrm{Tr}_E [U_t \rho_{SE} (0) U_t^\dagger] }[/math]

If we assume that the mapping [math]\displaystyle{ \rho_S(0) \mapsto \rho_S(t) }[/math] 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

[math]\displaystyle{ \rho_S = \sum_i F_i \rho_S (0) F_i^\dagger }[/math]

where the [math]\displaystyle{ F_i \, }[/math] 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 [math]\displaystyle{ \rho_{SE} (0) = \rho_S (0) \otimes \rho_E (0) }[/math], it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.

References

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