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

摘要

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