Uniqueness of Quantum Markov Chain Associated with XY-Ising Model on Cayley Tree of Order Two

Abstract

In this paper, we consider backward and forward Quantum Markov Chains (QMC) associated with XY -Ising model on the Cayley tree of order two. We construct finite volume states with boundary conditions, and define QMC as a weak limit of those states which depend on the boundary conditions. We prove that the limit state is a unique QMC associated with such a model, this means the QMC does not depend on the boundary conditions. Moreover, we observe the relation between backward and forward QMC.