Refinement of quantum Markov states on trees

TL;DR

This paper introduces a refined concept of quantum Markov states on trees, providing a structure theorem and characterizing translation-invariant states, advancing understanding of quantum stochastic processes on complex structures.

Contribution

It proposes a refinement of quantum Markov states on trees and proves a structure theorem, clarifying the properties of localized QMS and sub-Markov states.

Findings

Established a structure theorem for QMS on general trees

Identified that restrictions of QMS may not be QMS, but localized QMS are

Characterized translation-invariant QMS on regular trees

Abstract

In the present paper, we propose a refinement for the notion of quantum Markov states (QMS) on trees. A structure theorem for QMS on general trees is proved. We notice that any restriction of QMS in the sense of Ref. \cite{AccFid03} is not necessarily to be a QMS. It turns out that localized QMS has the mentioned property which is called \textit{sub-Markov states}, this allows us to characterize translation invariant QMS on regular trees.