Geodesic
Uniqueness of Quantum Markov Chains Associated with an $XY$-Model on a Cayley Tree of Order~$2$
Matematičeskie zametki, Tome 90 (2011) no. 2, pp. 168-182

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We propose the construction of a quantum Markov chain that corresponds to a “forward” quantum Markov chain. In the given construction, the quantum Markov chain is defined as the limit of finite-dimensional states depending on the boundary conditions. A similar construction is widely used in the definition of Gibbs states in classical statistical mechanics. Using this construction, we study the quantum Markov chain associated with an $XY$-model on a Cayley tree. For this model, within the framework of the given construction, we prove the uniqueness of the quantum Markov chain, i.e., we show that the state is independent of the boundary conditions.
Mots-clés : quantum Markov chain, phase transition, quasiconditional expectation
Keywords: Cayley tree, $XY$-model, Gibbs state, graph, dynamical system, quasilocal algebra. 
@article{MZM_2011_90_2_a1,
     author = {L. Accardi and F. M. Mukhamedov and M. Kh. Saburov},
     title = {Uniqueness of {Quantum} {Markov} {Chains} {Associated} with an $XY${-Model} on a {Cayley} {Tree} of {Order~}$2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {168--182},
     publisher = {mathdoc},
     volume = {90},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_2_a1/}
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L. Accardi; F. M. Mukhamedov; M. Kh. Saburov. Uniqueness of Quantum Markov Chains Associated with an $XY$-Model on a Cayley Tree of Order~$2$. Matematičeskie zametki, Tome 90 (2011) no. 2, pp. 168-182. http://geodesic.mathdoc.fr/item/MZM_2011_90_2_a1/