Belavkin-Staszewski Quantum Markov Chains

TL;DR

This paper introduces Belavkin-Staszewski quantum Markov chains by replacing the Umegaki entropy with BS-entropy, establishing a correspondence with quantum Markov chains, and exploring structural properties, recovery maps, and decay rates of conditional mutual information.

Contribution

It develops the theory of BS-quantum Markov chains, including a recovery map, structural decomposition, and analysis of approximate cases, extending the understanding of quantum Markov properties.

Findings

Established a correspondence between quantum and BS-quantum Markov chains.

Constructed a recovery map for BS-entropy similar to Petz recovery map.

Identified states with superexponentially decaying conditional mutual information.

Abstract

It is well-known that the conditional mutual information of a quantum state is zero if, and only if, the quantum state is a quantum Markov chain. Replacing the Umegaki relative entropy in the definition of the conditional mutual information by the Belavkin-Staszewski (BS) relative entropy, we obtain the BS-conditional mutual information, and we call the states with zero BS-conditional mutual information Belavkin-Staszewski quantum Markov chains. In this article, we establish a correspondence which relates quantum Markov chains and BS-quantum Markov chains. This correspondence allows us to find a recovery map for the BS-entropy in the spirit of the Petz recovery map. Furthermore, we show that, over the set of BS-quantum Markov chains, this correspondence constitutes an entanglement-breaking map. Moreover, we prove a structural decomposition of the Belavkin-Staszewski quantum Markov chains and also study states for which the BS-conditional mutual information is only approximately zero. We subsequently extend the aforementioned correspondence, structural decomposition and recovery map to arbitrary pairs of states and conditional expectations. As an application of the correspondence, we find the first family of states with non-vanishing conditional mutual information for which it decays superexponentially fast with the size of the middle system.