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. investigate property w.r.t. finer structure of considered tree. The main result present paper concerns diagonalizability locally faithful QMS $$\varphi $$ on UHF-algebra $${\mathcal {A}}_V$$ over by means suitable conditional expectation into maximal abelian subalgebra. Namely, we prove existence Umegaki $${\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {D}}_V$$ such that =\varphi _{\lceil {D}}_V}\circ {\mathfrak {E}}$$ . Moreover, it is clarified Markovian associated classical measure spectrum diagonal algebra