Operational Characterization of Divisibility of Dynamical Maps

TL;DR

This paper introduces an operational approach to characterizing the divisibility of quantum dynamical maps using quantum channel distinguishability, linking it to Markovianity and entropic measures.

Contribution

It provides a novel operational framework for assessing divisibility of quantum maps via channel distinguishability and entropic measures, advancing understanding of quantum Markovianity.

Findings

Distinguishability of quantum channels does not increase under divisible maps.

Operational characterization of $k$-divisibility using entanglement-assisted channel distinguishability.

Entropic characterization of divisible maps via min-entropy.

Abstract

Divisibility of dynamical maps turns out to be a fundamental notion in characterising Markovianity of quantum evolution, although the decision problem for divisibility itself is computationally intractable. In this work, we propose the operational characterisation of divisibility of dynamical maps by exploiting distinguishability of quantum channels. We prove that distinguishability for any pair of quantum channels does not increase under divisible maps, and then, in terms of channel distinguishability with entanglement between system and $k$-dimensional ancillas, provide the operational characterization for the full hierarchy of the so-called $k$-divisiblity $(k=1,2,\ldots)$. Finally, from the fact that min-entropy corresponds to the information-theoretic measure of distinguishability, the entropic characterisation to divisible maps is also provided.