Refinement of quantum Markov states on trees

Abstract

In the present paper, we propose a refinement for the notion of quantum Markov states (QMS) on trees. A structure theorem for QMS on general trees is proved. We notice that any restriction of QMS in the sense of reference Accardi and Fidaleo (2003 J. Funct. Anal. 200 324–347) is not necessarily to be a QMS. It turns out that localized QMS has the mentioned property which is called sub-Markov states, this allows us to characterize translation invariant QMS on regular trees.

Entropy of quantum Markov states on Cayley trees

“…However, the boundary condition plays a more physically significant role in the existence of phase transitions (see for instance [5,25,26,44]). In [28][29][30], the authors studied in detail the structure of QMSs associated with localized transitions expectations on the Cayley tree.…”

“…Furthermore, in [21,28,29] a description of QMSs has been carried out. It is worth stressing that the considered QMSs had the Markov property not only with respect to levels of the considered tree, but also with regard to the interaction domain at each site, which has a finer structure, and through a family of suitable quasiconditional expectations which are localized [28,30,32]. Such a localization property is essential for the integral decomposition of QMS, since it takes into account the finer structure of conditional expectations and filtration [2,9].…”