Quantum versus classical P-divisibility

P-divisibility is a central concept in both classical and quantum non-Markovian processes; in particular, it is strictly related to the notion of information backflow. When restricted to a fixed commutative algebra generated by a complete set of orthogonal projections, any quantum dynamics naturally provides a classical stochastic process. It is indeed well known that a quantum generator gives rise to a P-divisible quantum dynamics if and only if all its possible classical reductions give rise to divisible classical stochastic processes. However, this property does not hold if one operates a classical reduction of the quantum dynamical maps instead of their generators: As an example, for a unitary dynamics, P-divisibility of its classical reduction is inevitably lost and the latter thus exhibits information backflow. Instead, for some important classes of purely dissipative qubit evolutions, quantum P-divisibility always implies classical P-divisibility and therefore excludes information backflow in both the quantum and classical scenarios. On the contrary, for a wide class of orthogonally covariant qubit dynamics, we show that loss of classical P-divisibility originates from the classical reduction of a purely dissipative P-divisible quantum dynamics as in the unitary case. Moreover, such an effect can be interpreted in terms of information backflow due to the coherences developed by the quantumly evolving classical state.