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.