  Two long standing open problems in quantum theory are to characterize the
class of initial system-bath states for which quantum dynamics is equivalent to
(1) a map between the initial and final system states, and (2) a completely
positive (CP) map. The CP map problem is especially important, due to the
widespread use of such maps in quantum information processing and open quantum
systems theory. Here we settle both these questions by showing that the answer
to the first is "all", with the resulting map being Hermitian, and that the
answer to the second is that CP maps arise exclusively from the class of
separable states with vanishing quantum discord.
