Entropy of Quantum Markov states on Cayley trees

TL;DR

This paper advances the understanding of quantum Markov states on Cayley trees by defining and explicitly computing their mean entropies, addressing a key open problem in quantum probability and extending results to higher dimensions.

Contribution

It introduces a method to compute mean entropies of quantum Markov states on Cayley trees, significantly progressing in multi-dimensional quantum probability theory.

Findings

Explicit computation of mean entropies under certain conditions

Progress towards solving the open problem of entropy calculation in quantum Markov fields

Extension of quantum Markov state results from one dimension to multi-dimensional Cayley trees

Abstract

In this paper, we continue the investigation of quantum Markov states (QMS) and define their mean entropies. Such entropies are explicitly computed under certain conditions. The present work takes a huge leap forward at tackling one of the most important open problems in quantum probability which concerns the calculations of mean entropies of quantum Markov fields. Moreover, it opens new perspective for the generalization of many interesting results related to the one dimensional quantum Markov states and chains to multi-dimensional cases.