Bounds for the divisibility-based and distinguishability-based non-Markovianity measures

TL;DR

This paper establishes bounds for two non-Markovianity measures in quantum systems, providing exact values in some cases and illustrating their calculation with the spin-boson model.

Contribution

It derives an upper bound for the distinguishability-based measure and a lower bound for the divisibility-based measure, advancing understanding of non-Markovian quantum dynamics.

Findings

Exact measure values achieved for certain master equations.

Calculated bounds for spin-boson model examples.

Identified the role of drift vector in measure differences.

Abstract

We derive an upper bound for the distinguishability-based non-Markovianity measure of a two-level system and prove that for certain master equations the exact value of the measure achieves this bound. Furthermore, we obtain an easily calculable lower bound for the divisibility-based non-Markovianity measure of an $n$-level system. We illustrate the calculation of these bounds through examples, considering in detail the spin-boson model. We show that the differences between the two measures in the spin-boson model are caused by the drift vector that is also responsible for the non-unitality of the dynamical map.