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arXiv:math-ph/0411035

Markov states on the CAR algebra

Luigi Accardi, Francesco Fidaleo, Mukhamedov, Farruh

Published Nov 9, 2004• 1 citation
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Abstract

The program relative to the investigation of quantum Markov states for spin chains based on Canonical Anticommutation Relations algebra is carried on. This analysis provides a further step for a satisfactory theory of quantum Markov processes.
DOI
10.48550/arxiv.math-ph/0411035

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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.
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We present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti-commutation relations (CAR) in analogy to quasifree states on the algebra of canonical commutation relations (CCR). The used mathematical tools are based on a new algebraic structure the “Grassmann algebra of canonical anticommutation relations” (GAR algebra) which is given by the twisted tensor product of a Grassmann and a CAR algebra. As a new application, the corresponding theory provides an elegant tool for calculating the fidelity of two quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems.