Generalized trace distance measure connecting quantum and classical non-Markovianity

TL;DR

This paper introduces a generalized measure of quantum non-Markovianity based on Helstrom matrices, linking quantum and classical Markovianity through a criterion equivalent to P-divisibility, and demonstrates its properties with examples.

Contribution

It generalizes the trace distance measure of quantum non-Markovianity using Helstrom matrices, establishing a direct connection to classical Markovian processes and P-divisibility.

Findings

The measure is equivalent to P-divisibility of quantum processes.

Optimal states for maximal information backflow are orthogonal.

The generalized measure has a universal local representation.

Abstract

We establish a direct connection of quantum Markovianity of an open quantum system to its classical counterpart by generalizing the criterion based on the information flow. Here, the flow is characterized by the time evolution of Helstrom matrices, given by the weighted difference of statistical operators, under the action of the quantum dynamical evolution. It turns out that the introduced criterion is equivalent to P-divisibility of a quantum process, namely divisibility in terms of positive maps, which provides a direct connection to classical Markovian stochastic processes. Moreover, it is shown that similar mathematical representations as those found for the original trace distance based measure hold true for the associated, generalized measure for quantum non-Markovianity. That is, we prove orthogonality of optimal states showing a maximal information backflow and establish a local and universal representation of the measure. We illustrate some properties of the generalized criterion by means of examples.