Quantum Markov states on Cayley trees

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:473

Article

Mukhamedov, Farrukh¹  Souissi, Abdessatar^2,3 

[1] United Arab Emirates Univ, Coll Sci, Dept Math Sci, POB 15551, Abu Dhabi, U Arab Emirates

[2] Qassim Univ, Coll Business Adm, Buraydah, Saudi Arabia

[3] Carthage Univ, Preparatory Inst Sci & Tech Studies, Carthage, Tunisia

关键词: Quantum Markov state;    Localized;    Cayley tree;    Disintegration;    Ising type model;    Chain;   

DOI  :  10.1016/j.jmaa.2018.12.050

来源: Elsevier

【 摘 要 】

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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free