On Quantum Markov Chains on Cayley tree I: uniqueness of the associated chain with XY-model on the Cayley tree of order two

TL;DR

This paper constructs and analyzes quantum Markov chains on Cayley trees, specifically proving the uniqueness of the chain associated with the XY-model on a Cayley tree of order two, regardless of boundary conditions.

Contribution

It provides a new construction method for quantum Markov chains on Cayley trees and proves the uniqueness of the chain for the XY-model on a binary Cayley tree.

Findings

Uniqueness of the quantum Markov chain for the XY-model on a Cayley tree of order two.

Construction of states on finite volumes with boundary conditions leading to the QMC.

QMC does not depend on boundary conditions, indicating a unique state.

Abstract

In the present paper we study forward Quantum Markov Chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction we investigate QMC associated with XY-model on a Caylay tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions.