• DOI:10.1007/s10955-020-02674-1
• Corpus ID: 207780606

Diagonalizability of Quantum Markov States on Trees

@article{Mukhamedov2019DiagonalizabilityOQ,
  title={Diagonalizability of Quantum Markov States on Trees},
  author={Farrukh Mukhamedov and Abdessatar Souissi},
  journal={Journal of Statistical Physics},
  year={2019},
  volume={182},
  pages={1-15}
}

• F. Mukhamedov, A. Souissi
• Published 4 November 2019
• Mathematics
• Journal of Statistical Physics

We introduce quantum Markov states (QMS) in a general tree graph $$G= (V, E)$$ 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 $$\varphi $$ φ on a UHF-algebra $${\mathcal {A}}_V$$ 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 $${\mathfrak {E}} : {\mathcal {A}}_V \rightarrow {\mathcal {D}}_V$$ E : A V → D V such that $$\varphi =\varphi _{\lceil {\mathcal {D}}_V}\circ {\mathfrak {E}}$$ φ = φ ⌈ D V ∘ E . Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra $${\mathcal {D}}_V$$ D V .