Physically accessible non-completely positive maps

The Kraus representation theorem states that under some plausible assumptions, the most general dynamics that an open quantum system can undergo is a completely positive map (CP-map). However, one of the assumptions underlying the theorem is a condition on the initial state of the combined system and environment which is not always satisfied.We will describe the physical reasons why this assumption may not be true and give a natural definition for when it makes sense to say that a non-CP map is robustly physically realizable (i.e., physically accessible). We will also discuss what is known about the conditions for a non-CP map to be physically accessible.