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The effect of initial correlations on the evolution of quantum states

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Until fairly recently, it
was generally assumed that the initial state of a quantum system prepared for information processing was in a product state with its environment.  If this is the case,

the evolution is described by a completely positive map.  However, if the system and environment are initially correlated, or entangled, such that the so-called quantum discord is non-zero, then the

evolution is described by a map which is not completely positive. Maps that are not completely positive are not as well understood and the implications of having such a map are not completely known.  I will discuss a few examples and a theorem (or two) which may help us understand the implications of having maps which are not completely positive.