Hierarchy of Non-Markovianity and $k$-divisibility phase diagram of Quantum Processes in Open Systems

TL;DR

This paper introduces a $k$-divisibility phase diagram to classify quantum non-Markovianity, clarifying discrepancies between existing measures and providing deeper insight into quantum processes in open systems.

Contribution

It develops a $k$-divisibility phase diagram using $k$-positive maps, offering a new framework to understand and compare quantum non-Markovianity measures.

Findings

Explains the origin of discrepancies between common non-Markovianity measures.

Identifies conditions where different measures of non-Markovianity agree.

Provides a unified phase diagram to classify quantum processes based on $k$-divisibility.

Abstract

In recent years, much effort has been devoted to the construction of a proper measure of quantum non-Markovianity. However, those proposed measures are shown to be at variance with different situations. In this work, we utilize the theory of $k$-positive maps to generalize a hierarchy of $k$-divisibility and develop a powerful tool, called $k$-divisibility phase diagram, which can provide a further insight into the nature of quantum non-Markovianity. By exploring the phase diagram with several paradigms, we can explain the origin of the discrepancy between two frequently used measures and find the condition under which the two measures coincide with each other.