Narrow escape problem

2013-12-07T23:32:43Z

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In [[quantum mechanics]], especially in the study of [[open quantum system]]s, '''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>\rho_{SE} (0) \,</math> (which in general may be [[quantum entanglement|entangled]]) and undergoing unitary evolution given by <math>U_t \,</math>. Then the reduced dynamics of the system alone is simply
:<math>\rho_S (t) = \mathrm{Tr}_E [U_t \rho_{SE} (0) U_t^\dagger] </math>
If we assume that the mapping <math>\rho_S(0) \mapsto \rho_S(t)</math> is [[linear map|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>\rho_S = \sum_i F_i \rho_S (0) F_i^\dagger </math>
where the <math>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>\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.<ref>P. Pechukas, Reduced Dynamics Need Not Be Completely Positive. ''Physical Review Letters'' '''73''', 1060 (1994).</ref>

== Notes ==
<references/>

==References==
* Nielsen, Michael A. and [[Isaac L. Chuang]] (2000). ''Quantum Computation and Quantum Information'', Cambridge University Press, ISBN 0-521-63503-9

[[Category:Quantum information science]]

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Narrow escape problem