Diagonalizability of quantum Markov States on trees

We introduce quantum Markov states (QMS) in a general tree graph $G= (V, E)$, extending the Cayley trees 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 $mathcal D_V$.