Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Infinite dimensional entangled Markov chains

Infinite dimensional entangled Markov chains We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Random Operators and Stochastic Equations de Gruyter

Infinite dimensional entangled Markov chains

Loading next page...
 
/lp/de-gruyter/infinite-dimensional-entangled-markov-chains-0QKF1UZsgM

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
de Gruyter
Copyright
Copyright 2003, Walter de Gruyter
ISSN
0926-6364
eISSN
1569-397x
DOI
10.1515/1569397042722328
Publisher site
See Article on Publisher Site

Abstract

We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state–space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on , F being any infinite dimensional type I factor, J a finite interval of , and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a rôle in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type II 1 von Neumann factors.

Journal

Random Operators and Stochastic Equationsde Gruyter

Published: Dec 1, 2004

Keywords: Non commutative measure,; integration and probability;; Classifications of C ∗ – algebras,; factors;; Applications of selfadjoint operator algebras to physics;; Quantum Markov processes;; Quantum random walks;; Quantum information theory.

There are no references for this article.