Article
Version 1
Preserved in Portico This version is not peer-reviewed
Logical Entropy: Introduction to Classical and Quantum Logical Information Theory
Version 1
: Received: 10 August 2018 / Approved: 10 August 2018 / Online: 10 August 2018 (15:53:51 CEST)
A peer-reviewed article of this Preprint also exists.
Ellerman, D. Logical Entropy: Introduction to Classical and Quantum Logical Information Theory. Entropy 2018, 20, 679. Ellerman, D. Logical Entropy: Introduction to Classical and Quantum Logical Information Theory. Entropy 2018, 20, 679.
Abstract
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 (`dits') of a partition (a pair of points distinguished by the 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 (cohered together in the pure superposition state being measured) that are distinguished by the measurement (decohered in the post-measurement mixed state). 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.
Keywords
logical entropy, logic of partitions, qudits of an observable
Subject
Computer Science and Mathematics, Computer Science
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment