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 Q...
Ausführliche Beschreibung

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 %${\mathcal {D}}_V%$. Ausführliche Beschreibung