The Okinawa Lectures on Entropy

[2608.14523] The Okinawa Lectures on Entropy Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematical Physics arXiv:2608.14523 (math-ph) [Submitted on 14 Aug 2026] Title: The Okinawa Lectures on Entropy Authors: Klaas Landsman View a PDF of the paper titled The Okinawa Lectures on Entropy, by Klaas Landsman View PDF HTML (experimental) Abstract: After a historical introduction, the most important classical and quantum entropies are introduced as constructions in classical and quantum probability theory. Classical entropies are studied from large deviation theory, including theorems of Sanov, Cramér, Gärtner-Ellis, and Varadhan, and are illustrated in some applications to both Boltzmannian and Gibbsian statistical physics. Quantum entropies superficially connect to classical entropies at the formula level, but more deeply do so via the crucial role of entropy in statistical hypothesis testing. The classical (relative) Kullback-Leibler entropy, its quantum counterpart introduced by Umegaki, as well as their deformations proposed by Renyi all fit naturally in this context. Quantum entropy faces the new problem of defining and computing the relative entropy of a pair of states on a subsystem, here formalized as a von Neumann algebra. This requires modular (aka Tomita-Takesaki) theory, which provides the framework for the relative quantum entropies introduced by Araki and Uhlmann (these encompass both the Kullback-Leibler and Umegaki entropies as special cases). To (re)define and compute these entropies in terms of density operators and traces, further constructions are needed, namely Haagerup's noncommutative L^p spaces. Von Neumann algebras also provide the setting for the Connes-Stormer-Narnhofer-Thirring entropy, which is a quantum version of the Kolmogorov-Sinai entropy in dynamical systems and ergodic theory, to which we also provide an introduction. This course was originally inspired by, and should be relevant to, black hole thermodynamics, although we discuss neither this application nor the second law. The course tries to be both mathematically rigorous and interesting to theoretical physicists. Prerequisites are undergraduate probability theory, functional analysis, and quantum theory. Comments: 259 pages, 8 figures Subjects: Mathematical Physics (math-ph) ; General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th) MSC classes: 81P17, 60F10 (primary), 46L60, 80-02, 94A17 (secondary) Cite as: arXiv:2608.14523 [math-ph] (or arXiv:2608.14523v1 [math-ph] for this version) https://doi.org/10.48550/arXiv.2608.14523 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Klaas Landsman [ view email ] [v1] Fri, 14 Aug 2026 17:43:39 UTC (1,970 KB) Full-text links: Access Paper: View a PDF of the paper titled The Okinawa Lectures on Entropy, by Klaas Landsman View PDF HTML (experimental) TeX Source view license Current browse context: math-ph < prev | next > new | recent | 2026-08 Change to browse by: gr-qc hep-th math math.MP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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