Diagonalizability of quantum Markov States on trees

TL;DR

This paper extends quantum Markov states to general tree graphs, proving their diagonalizability via conditional expectations and clarifying the classical Markov structure on the spectrum of the diagonal algebra.

Contribution

It introduces the diagonalization of quantum Markov states on trees using conditional expectations, generalizing previous results beyond Cayley trees.

Findings

Existence of a conditional expectation diagonalizing QMS

Representation of QMS as a classical measure on the spectrum

Clarification of the Markovian structure of the classical measure

Abstract

We introduce quantum Markov states (QMS) in a general tree graph $G= (V, E)$, extending the Cayley tree's case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of this paper concerns the diagonalizability of a locally faithful QMS $\varphi$ on a UHF-algebra $\mathcal A_V$ over the considered tree by means of a suitable conditional expectation into a maximal abelian subalgebra. Namely, we prove the existence of a Umegaki conditional expectation $\mathfrak E : \mathcal A_V \to \mathcal D_V$ such that $$\varphi = \varphi_{\lceil \mathcal D_V}\circ \mathfrak E.$$ Moreover, we clarify the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra $\mathcal D_V$.