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

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.

19 pages

References in corpus (2)

• Quantum Markov fields on graphs
• Diagonalizability of non homogeneous quantum Markov states and associated von Neumann algebras

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• Quantum Markov Chains on the Comb graphs: Ising model
• Entropy of Quantum Markov states on Cayley trees
• The entropy and reversibility of cellular automata on Cayley tree
• A Forward Quantum Markov Field on Graphs