Advances in Operator Theory

Construction of a new class of quantum Markov fields

Luigi Accardi, Farrukh Mukhamedov, and Abdessatar Souissi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, it allows us to express the Markov property for the local structure of the graph. Our main result is the existence and uniqueness of quantum Markov field.

Article information

Adv. Oper. Theory, Volume 1, Number 2 (2016), 206-218.

Received: 13 October 2016
Accepted: 14 December 2016
First available in Project Euclid: 4 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L53: Noncommutative probability and statistics
Secondary: 60J99, 46L60, 60G50, 82B10

quantum Markov field graph tessellation construction


Accardi, Luigi; Mukhamedov, Farrukh; Souissi, Abdessatar. Construction of a new class of quantum Markov fields. Adv. Oper. Theory 1 (2016), no. 2, 206--218. doi:10.22034/aot.1610.1031.

Export citation


  • L. Accardi, The noncommutative Markov property, (Russian) Funkcional. Anal. i Priložen. 9 (1975), no. 1, 1–8.
  • L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a Theorem of Takesaki, J. Funct. Anal. 45 (1982), no. 2, 245–273.
  • L. Accardi and F. Fidaleo, Quantum Markov fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 1, 123–138.
  • R. L. Dobrushin, Description of Gibbsian Random Fields by means of conditional probabilities, Probab. Theory Appl. 13 (1968), no. 2, 201–229.
  • L. Accardi, H. Ohno, and F. Mukhamedov, Quantum Markov fields on graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 2, 165–189.
  • M. Fannes, B. Nachtergaele, and R. F. Werner, Ground states of VBS models on Cayley trees, J. Statist. Phys. 66 (1992), no. 3-4, 939–973.
  • M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Commun. Math. Phys. 144 (1992), no. 3, 443–490.
  • H.-O. Georgi, Gibbs measures and phase transitions, de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 1988.
  • F. Mukhamedov, A. Barhoumi, and A. Souissi, Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree, J. Stat. Phys.163 (2016), no. 3, 544–567.
  • V. Liebscher, Markovianity of quantum random fields, Proceedings Burg Conference 15–20 March 2001, W. Freudenberg (ed.), World Scientific, QP–PQ Series 15 (2003), 151–159.
  • C. Preston, Gibbs states on countable sets, Cambridge Tracts in Mathematics, No. 68. Cambridge University Press, London-New York, 1974.
  • A. Spataru, Construction of a Markov field on an infinite tree, Advance Math. 81 (1990), no. 1, 105–116.
  • S. Zachary, Countable state space Markov random fields and Markov chains on trees, Ann. Prob. 11 (1983),no. 4, 894–903.