Quantum Information & Computation
Volume 25, Issue 1
RESEARCH ARTICLE Open access

A New Logical Measure for Quantum Information

David Ellerman

Author affiliations
  • David Ellerman1
  1. Independent Researcher, Ljubljana, Slovenia

pp. 82-96 · 10.2478/qic-2025-0005

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Abstract

Starting at the logical level of the logic of partitions, dual to the usual Boolean logic of subsets, the notion of logical entropy, i.e., information as distinctions, is developed as the quantification of the distinctions of partitions—just as probability theory starts with the quantification of elements of subsets. Logical entropy is compared and contrasted with the usual notion of Shannon entropy. Then a semi-algorithmic procedure (from the mathematical folklore) is used to translate the notion of logical entropy at the set level to the corresponding notion of quantum logical entropy at the (Hilbert) vector space level. Examples and some important propositions relate quantum logical entropy to projective measurement and to the Hilbert-Schmidt distance. Overall, the approach demonstrates the logical basis and naturality of this notion of entropy for quantum mechanics.
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Keywords

  • partition logic
  • duality of subsets and partitions
  • logical entropy
  • Shannon entropy
  • linearization of set concepts to vector space concepts
  • quantum logical entropy

Suggested citation

David Ellerman (2025). A New Logical Measure for Quantum Information. Quantum Information & Computation, 25(1), 82-96. https://doi.org/10.2478/qic-2025-0005

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