Quantum vs. classical P-divisibility

It is well known that a quantum generator gives rise to a P-divisible quantum dynamics if and only if all its possible classical reductions to a commutative algebra generate P-divisible classical stochastic processes. Yet, this property does not hold if one classically reduces the quantum dynamical maps instead of their generators, as inevitably occurs, for instance, for unitary dynamics. On the other hand, for some important classes of purely dissipative evolutions, quantum P-divisibility always implies classical P-divisibility. We provide an interpretation of the loss of classical P-divisibility in terms of information flow, the information coming in being stored in the coherences of the time-evolving quantum state. As in the unitary case, we show that the loss of classical P-divisibility can also originate from the classical reduction of a purely dissipative P-divisible quantum dynamics.