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Infinite Dimensional Analysis Quantum Probability and Related Topics

MARKOV STATES AND CHAINS ON THE CAR ALGEBRA

Luigi Accardi, Francesco Fidaleo, Farrukh Maksutovich Mukhamedov

Published Jun 1, 2007• 45 citations
Publisher

Abstract

We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over ℤ, emphasizing some remarkable differences with the infinite tensor product case. We describe the structure of the Markov states on this algebra and show that, contrarily to the infinite tensor product case, not all these states are diagonalizable. A general method to construct nontrivial quantum Markov chains on the CAR algebra is also proposed and illustrated by some pivotal examples. This analysis provides a further step for a satisfactory theory of quantum Markov processes.

DOI
10.1142/s0219025707002683

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We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over ℤ, emphasizing some remarkable differences with the infinite tensor product case. We describe the structure of the Markov states on this algebra and show that, contrarily to the infinite tensor product case, not all these states are diagonalizable. A general method to construct nontrivial quantum Markov chains on the CAR algebra is also proposed and illustrated by some pivotal examples. This analysis provides a further step for a satisfactory theory of quantum Markov processes.
We investigate the structure of the Markov states on general Fermion algebras. The situation treated in the present paper covers, beyond the d--Markov states on the CAR algebra on Z (i.e. when there are d--annihilators and creators on each site), also the non homogeneous case (i.e. when the numbers of generators depends on the localization). The present analysis provides the first necessary step for the study of the general properties, and the construction of nontrivial examples of Fermi Markov states on the d--standard lattice, that is the Fermi Markov fields. Natural connections with the KMS boundary condition and entropy of Fermi Markov states are studied in detail. Apart from a class of Markov states quite similar to those arising in the tensor product algebras (called "strongly even" in the sequel), other interesting examples of Fermi Markov states naturally appear. Contrarily to the strongly even examples, the latter are highly entangled and it is expected that they describe interactions which are not "commuting nearest neighbor". Therefore, the non strongly even Markov states, in addition to the natural applications to quantum statistical mechanics, might be of interest for the information theory as well.
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.
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.
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 ∪J⊂Z⊗ˉJFˉC∗\bar{\cup_{J\subset Z} \bar{\otimes}_{J}F}^{C^{*}}∪J⊂Z​⊗ˉ​J​Fˉ​C∗, FFF being any infinite dimensional type III factor, JJJ a finite interval of ZZZ, 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 II1II_1II1​ von Neumann factors.
Abstract We deal with the analogous of the so-called Quantum Markov Chains for Fermi systems (i.e. the models whose fundamental fields enjoy the Canonical Anti-commutation Relations), and discuss their main properties. Since the involution and the product highly differ from those of the usual tensor product, the positivity of the involved operators and functionals, not automatically satisfied, is a very delicate question. As a consequence, the positivity should be checked directly for the specific models under consideration. Therefore, we analyse some pivotal cases, providing examples which go beyond the simpler ones already studied in literature. As an interesting result, we prove that the symmetric states, that is those which are invariant under the natural action of the group of all finite permutations, are described in terms of particular Quantum Markov Chains.