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 ${\rho }_{SE}(0)$ (which in general may be entangled) and undergoing unitary evolution given by $U_t$. Then the reduced dynamics of the system alone is simply

${\rho }_S(t)={\mathrm{T}\mathrm{r}}_E[U_t{\rho }_{SE}(0)U_t^{†}]$

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

${\rho }_S=\sum _i{F}_i{\rho }_S(0)F_i^{†}$

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 ${\rho }_{SE}(0)={\rho }_S(0)\otimes {\rho }_E(0)$, it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.

References

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