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.Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

Diagonalizability of quantum Markov States on trees

“…The proof uses the same argument used in [29] where a similar result has been established in the case of the Cayley trees.…”

“…The reader is referred to [7,17,18,21] for recent development of the theory and its applications On the other hand, in [25,26] we have investigated a quantum Markov chains associated to a particular class of the Ising models with competing (commuting) interactions on the Cayley trees. Recently, in [29] we have established that the above considered QMCs define a special class called Quantum Markov States (QMS). Furthermore, description of QMS has been carried out.…”

Quantum Markov Chains on the Comb graphs: Ising model

“…One can see that the localized transition expectation E [n,n+1] highlights the fine structure of the considered graph. This property played an essential role in the study of phase transitions for quantum Markov chains on the Cayley tree [37].…”

“…In this path the QMC scheme is based on the C * -algebraic framework (see also [10]). Furthermore, in [29,30,31,35,36,37] we have established that Gibbs measures of the Ising model with competing (Ising) interactions (with commuting interactions) on a Cayley trees, can be considered as QMC. Note that if the perturbation vanishes then the model reduces to the classical Ising one which was also examined in [14] by means of C * -algebraic methods.…”

Entropy of Quantum Markov states on Cayley trees

“…We point out that, in [6,27,28,29] we have started to investigate particular classes of quantum Markov chains (QMC) associated to the Ising types models on the Cayley trees. It turned out that the above considered QMCs fall into a special class called quantum Markov states (QMS) (see [30,31]). Furthermore, in [22,30,31] a description of QMS has been carried out.…”