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Open Physics

Logical entropy of quantum dynamical systems

Abolfazl Ebrahimzadeh

Published Jan 1, 2016• 16 citations
Publisher

Abstract

Abstract This paper introduces the concepts of logical entropy and conditional logical entropy of hnite partitions on a quantum logic. Some of their ergodic properties are presented. Also logical entropy of a quantum dynamical system is dehned and ergodic properties of dynamical systems on a quantum logic are investigated. Finally, the version of Kolmogorov-Sinai theorem is proved.

DOI
10.1515/phys-2015-0058

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Abstract This paper introduces the concepts of logical entropy and conditional logical entropy of hnite partitions on a quantum logic. Some of their ergodic properties are presented. Also logical entropy of a quantum dynamical system is dehned and ergodic properties of dynamical systems on a quantum logic are investigated. Finally, the version of Kolmogorov-Sinai theorem is proved.
‎This contribution ‎‎deals with the mathematical modeling of R-norm entropy and R-norm divergence in quantum logics. ‎‎W‎e extend some results concerning the R-norm entropy and conditional R-norm entropy given in (Inf. Control 45, 1980), ‎‎‎‎‎to ‎the‎ quantum logics.‎ Firstly, the concepts of ‎‎R-norm entropy ‎and‎ ‎conditional R-norm entropy in quantum logics are introduced. ‎We ‎prove‎ ‎the concavity property for the notion of R-norm entropy in quantum logics ‎and we ‎show‎‎ that this entropy measure does not have the property of sub-additivity in a true sense. ‎It ‎is ‎prove‎n that ‎‎‎the monotonicity ‎property ‎for ‎the suggested type of ‎conditional ‎version ‎of ‎R-norm ‎entropy, holds. Furthermore, we introduce the concept of R-norm divergence of states in quantum logics and we derive basic properties of this quantity. ‎In particular‎, a relationship between the R-norm divergence and the R-norm entropy of partitions is provided.‎‎‎
Logical entropy gives a measure, in the sense of measure theory, of the distinctions of a given partition of a set, an idea that can be naturally generalized to classical probability distributions. Here, we analyze how this fundamental concept and other related definitions can be applied to the study of quantum systems with the use of quantum logical entropy. Moreover, we prove several properties of this entropy for generic density matrices that may be relevant to various areas of quantum mechanics and quantum information. Furthermore, we extend the notion of quantum logical entropy to post-selected systems.
Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions of a partition. All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates that are distinguished by the measurement. Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states.
Several quantum dynamical entropies have been proposed that extend the classical Kolmogorov–Sinai (dynamical) entropy. The same scenario appears in relation to the extension of algorithmic complexity theory to the quantum realm. A theorem of Brudno establishes that the complexity per unit time step along typical trajectories of a classical ergodic system equals the KS-entropy. In the following, we establish a similar relation between the Connes–Narnhofer–Thirring quantum dynamical entropy for the shift on quantum spin chains and the Gács algorithmic entropy. We further provide, for the same system, a weaker linkage between the latter algorithmic complexity and a different quantum dynamical entropy proposed by Alicki and Fannes.