Diagonalizability of quantum Markov States on trees

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 $\phi$ 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 $\mathfrak E : \mathcal A_V \to \mathcal D_V$suchthat$$\phi ={\phi }_{\lceil {\mathcal{D}}_V}\circ \mathfrak{E}\mathrm{.}$$ Moreover, we clarify the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra ${\mathcal{D}}_V$.