Quantum versus classical $P$-divisibility

TL;DR

This paper investigates the relationship between quantum and classical $P$-divisibility, revealing conditions under which classical reductions reflect quantum properties and how information backflow manifests in different quantum dynamics.

Contribution

It clarifies when classical reductions of quantum dynamics preserve $P$-divisibility and how information backflow relates to quantum coherences in various qubit evolutions.

Findings

Quantum $P$-divisibility implies classical $P$-divisibility in certain dissipative qubit evolutions.

Loss of classical $P$-divisibility can originate from quantum dynamics, indicating information backflow.

Classical reductions may not always reflect quantum divisibility properties, especially in unitary cases.

Abstract

$P$-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. Yet, 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, which thus exhibits information backflow. Instead, for some important classes of purely dissipative qubit evolutions, quantum $P$-divisibility always implies classical $P$-divisibility and thus lack of information backflow both in 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 can originate 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, the information coming in being stored in the coherences of the time-evolving quantum state.