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

${\displaystyle \rho _{S}(t)=\mathrm {Tr} _{E}}$

If we assume that the mapping ${\displaystyle \rho _{S}(0)\mapsto \rho _{S}(t)}$ 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

${\displaystyle \rho _{S}=\sum _{i}F_{i}\rho _{S}(0)F_{i}^{\dagger }}$

where the ${\displaystyle F_{i}\,}$ 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 ${\displaystyle \rho _{SE}(0)=\rho _{S}(0)\otimes \rho _{E}(0)}$, it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.

Notes

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

References

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

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