## Abstract

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.

Original language | English |
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Pages (from-to) | 162-174 |

Number of pages | 13 |

Journal | Mathematical Notes |

Volume | 90 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 2011 |

Externally published | Yes |

## Keywords

- Cayley tree
- Gibbs state
- XY-model
- dynamical system
- graph
- phase transition
- quantum Markov chain
- quasiconditional expectation
- quasilocal algebra

## ASJC Scopus subject areas

- Mathematics(all)