  Complete positivity of quantum dynamics is often viewed as a litmus test for
physicality, yet it is well known that correlated initial states need not give
rise to completely positive evolutions. This observation spurred numerous
investigations over the past two decades attempting to identify necessary and
sufficient conditions for complete positivity. Here we describe a complete and
consistent mathematical framework for the discussion and analysis of complete
positivity for correlated initial states of open quantum systems. This
formalism is built upon a few simple axioms and is sufficiently general to
contain all prior methodologies going back to Pechakas, PRL (1994). The key
observation is that initial system-bath states with the same reduced state on
the system must evolve under all admissible unitary operators to system-bath
states with the same reduced state on the system, in order to ensure that the
induced dynamical maps on the system are well-defined. Once this consistency
condition is imposed, related concepts like the assignment map and the
dynamical maps are uniquely defined. In general, the dynamical maps may not be
applied to arbitrary system states, but only to those in an appropriately
defined physical domain. We show that the constrained nature of the problem
gives rise to not one but three inequivalent types of complete positivity.
Using this framework we elucidate the limitations of recent attempts to provide
conditions for complete positivity using quantum discord and the quantum
data-processing inequality. The problem remains open, and may require fresh
perspectives and new mathematical tools. The formalism presented herein may be
one step in that direction.
