# Quantum Markov States on Cayley trees

## TL;DR

This paper characterizes quantum Markov states on Cayley trees, showing they can be represented as integrals over product states and as Gibbs states with commuting interactions, extending understanding beyond one-dimensional cases.

## Contribution

It provides the first characterization of quantum Markov states on Cayley trees, linking them to Gibbs states with commuting interactions and integral representations.

## Key findings

- QMS on Cayley trees can be realized as integrals of product states.
- Locally faithful QMS correspond to Gibbs states with commuting interactions.
- Results extend the understanding of QMS beyond one-dimensional models.

## 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.