Quantum Markov states on Cayley trees

Farrukh Mukhamedov, Abdessatar Souissi

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)


It is known that any locally faithful quantum Markov state (QMS) on one dimensional setting can be considered as a Gibbs state associated with Hamiltonian with commuting nearest-neighbor interactions. In our previous results, we have investigated quantum Markov states (QMS) associated with Ising type models with competing interactions, which are expected to be QMS, but up to now, there is no any characterization of QMS over trees. We notice that these QMS do not have one-dimensional analogues, hence results of related to one dimensional QMS are not applicable. Therefore, the main aim of the present paper is to describe of QMS over Cayley trees. Namely, we prove that any QMS (associated with localized conditional expectations) can be realized as integral of product states w.t.r. a Gibbs measure. Moreover, it is established that any locally faithful QMS associated with localized conditional expectations can be considered as a Gibbs state corresponding to Hamiltonians (on the Cayley tree) with commuting competing interactions.

Original languageEnglish
Pages (from-to)313-333
Number of pages21
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - May 1 2019


  • Cayley tree
  • Chain
  • Disintegration
  • Ising type model
  • Localized
  • Quantum Markov state

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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