# Journal of Operator Theory

Volume 66, Issue 2, Fall 2011 pp. 385-414.

Fermi-Markov states**Authors**: Francesco Fidaleo

**Author institution:**Department of Mathematics, University of Tor Vergata, Rome, 00133, Italy

**Summary:**We investigate the structure of the Markov states on general\break Fermi algebras. The situation treated in the present paper covers, beyond the $d$-Markov states on the CAR algebra on $\bz$ (i.e.\ when there are $d$ Fermionic annihilators and creators on each site), also the nonhomogeneous 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 $\bz^{\n}$, 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 quantum information theory as well.

**Keywords:**selfadjoint operator algebras, quantum Markov processes, mathematical quantum statistical mechanics

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