Diagonalizability of Quantum Markov States on Trees

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 the present 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 $${\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {\mathcal {D}}_V$$ such that $$\varphi =\varphi _{\lceil {\mathcal {D}}_V}\circ {\mathfrak {E}}$$ . Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra $${\mathcal {D}}_V$$ .