Phase space measures of information flow in open systems: A quantum and classical perspective of non-Markovianity

TL;DR

This paper explores the relationship between quantum and classical measures of information flow in open systems, demonstrating that the Wigner function distance effectively quantifies non-Markovianity and the quantum-to-classical transition.

Contribution

It establishes a connection between trace distance and Kolmogorov distance for phase space distributions, introducing the Wigner function distance as a tool for measuring non-Markovianity.

Findings

Kolmogorov distance can match trace distance for certain quasi-probability distributions.

Wigner function distance converges rapidly to trace distance in the classical limit.

Wigner function distance is effective for semi-classical information backflow measurement.

Abstract

The exchange of information between an open quantum system and its environment, especially the backflow of information from the environment to the open system associated with quantum notions of non-Markovianity, is a widely discussed topic for years now. This information flow can be quantified by means of the trace distance of pairs of quantum states which provides a measure for the distinguishability of the states. The same idea can also be used to characterize the information flow in classical open systems through a suitable distance measure for their probability distributions on phase space. Here, we investigate the connection between the trace distance based quantum measure and the Kolmogorov distance for differently ordered quasi-probability distributions on phase space. In particular, we show that for any pair of quantum states one can find a unique quasi-probability distribution for which the Kolmogorov distance coincides with the trace distance. We further study the quantum-to-classical transition of the distance measures. Employing the Caldeira-Legget model of quantum Brownian motion as a prototypical example, numerical simulations indicate a particularly rapid convergence of the Kolmogorov distance of the Wigner functions to the trace distance in the classical uncertainty limit, which establishes the Wigner function distance as an optimal tool for measuring semi-classical information backflow and for quantifying non-Markovianity in open continuous variable quantum systems.