Open system quantum evolution and the assumption of complete positivity (A Tutorial)

Quantum systems which interact with an unknown environment cannot be described in terms of a unitary evolution on the system alone. For such evolution one can use a map from one density operator to another and use any other known information to model the system. Such maps are required to be positive (at least on their domain) - they take positive density operators to positive density operators - so as to be physically reasonable. A map Φ which is positive but not completely positive (CP) is one which takes positive operators to positive operators, but when extended by the identity In, i.e. I_n I_n⊗ Φ does not give a positive map for some In. The map is CP if and only if the extension is positive for all n. Recently some effort has been put forth to try to understand if, and/or, under what circumstances one might utilize a non-CP map. Here, after a tutorial-type introduction to maps from one density operator to another, some examples which do not fit the standard prescription (SP) for deriving a CP map are given. In cases where the physical system is constrained by some knowledge of the interaction, it is possible that the SP cannot be used directly. Our example is robust to changes in the initial and final conditions.