Another subset of collaborators and I have posted a new paper in the arXiv:
Dynamical maps can be nonpositive due to the constraints imposed by the initial system-environment correlation. We find the conditions for positivity and complete positivity of such dynamical maps by using the assignment map. Any initial system-environment correlations make the assignment map nonpositive, while the positivity of the dynamical depends on the interplay between the assignment map and the system-environment coupling. We show how this interplay can reveal and/or hide the nonpositivity of the assignment map. We discuss how this is related to many Markovian models.
This paper was done in collaboration with Kavan Modi and Alán Aspuru-Guzik.