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 the present paper concerns the diagonalizability of a locally faithful QMS φ on a UHF-algebra 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 E: A_V→ D_V such that φ=φ⌈DV∘E. Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra D_V.