On quantum markov chains on cayley tree i: Uniqueness of the associated chain with xy-model on the cayley tree of order two

Luigi Accardi; Farrukh Mukhamedov; Mansoor Saburov

doi:10.1142/S021902571100447X

On quantum markov chains on cayley tree i: Uniqueness of the associated chain with xy-model on the cayley tree of order two

Luigi Accardi
, Farrukh Mukhamedov
, Mansoor Saburov

Research output: Contribution to journal › Article › peer-review

28 Citations (Scopus)

Abstract

In this 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 Cayley 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.

Original language	English
Pages (from-to)	443-463
Number of pages	21
Journal	Infinite Dimensional Analysis, Quantum Probability and Related Topics
Volume	14
Issue number	3
DOIs	https://doi.org/10.1142/S021902571100447X
Publication status	Published - Sept 2011
Externally published	Yes

Keywords

Cayley tree
Quantum Markov chain
XY-model
uniqueness

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Mathematical Physics
Applied Mathematics

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10.1142/S021902571100447X

Cite this

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title = "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 this 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 Cayley 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.",

keywords = "Cayley tree, Quantum Markov chain, XY-model, uniqueness",

author = "Luigi Accardi and Farrukh Mukhamedov and Mansoor Saburov",

note = "Funding Information: The present study have been done within the Grant FRGS0308-91 of Malaysian Ministry of Higher Education. A final part of this work was done at the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy. F.M. thanks the ICTP for providing financial support of his visit (within the scheme of Junior Associate) to ICTP. The authors (F.M., M.S.) also acknowledge the Malaysian Ministry of Science, Technology and Innovation Grant 01-01-08-SF0079.",

year = "2011",

month = sep,

doi = "10.1142/S021902571100447X",

language = "English",

volume = "14",

pages = "443--463",

journal = "Infinite Dimensional Analysis, Quantum Probability and Related Topics",

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AU - Accardi, Luigi

AU - Mukhamedov, Farrukh

AU - Saburov, Mansoor

N1 - Funding Information: The present study have been done within the Grant FRGS0308-91 of Malaysian Ministry of Higher Education. A final part of this work was done at the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy. F.M. thanks the ICTP for providing financial support of his visit (within the scheme of Junior Associate) to ICTP. The authors (F.M., M.S.) also acknowledge the Malaysian Ministry of Science, Technology and Innovation Grant 01-01-08-SF0079.

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N2 - In this 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 Cayley 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.

AB - In this 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 Cayley 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.

KW - Cayley tree

KW - Quantum Markov chain

KW - XY-model

KW - uniqueness

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UR - https://www.scopus.com/pages/publications/79961030843#tab=citedBy

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SN - 0219-0257

VL - 14

SP - 443

EP - 463

JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics

JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics

IS - 3

ER -

On quantum markov chains on cayley tree i: Uniqueness of the associated chain with xy-model on the cayley tree of order two

Abstract

Keywords

ASJC Scopus subject areas

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