Quantum Markov States on Cayley trees

Abstract

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.

21 Pages

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Cited by in corpus (5)

• Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley tree
• Open Quantum Random Walks and Quantum Markov chains on Trees I: Phase transitions
• Refinement of quantum Markov states on trees
• Quantum Markov Chains on the Comb graphs: Ising model
• Entropy of Quantum Markov states on Cayley trees