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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    I have added

    • Paolo Facchi, Giovanni Gramegna, Arturo Konderak, Entropy of quantum states (arXiv:2104.12611)

    and added publication details to

    • A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Algebraic approach to entanglement and entropy, Phys. Rev. A 88, 022301 (2013) (arXiv:1301.1300)

    and grouped together more discernibly the references on operator-algebraic entropy

    diff, v50, current

  2. Fixed a broken link in the references.

    Rongmin Lu

    diff, v55, current

  3. Added details to a reference.

    Rongmin Lu

    diff, v57, current

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeAug 15th 2022
    Although the formula for the entropy of an Algebra is correct (or at least consistent with the definition of Peter Walters "An introduction to Ergodic Theory" ),
    subsequent link to density distribution entropy is incorrect.
    Indeed, one can readily check that the entropy for any density distribution computed with this definition is infinite.
    One can also see that entropy with this definition is an invariant of the measured sigma algebra. It thus yield the same value for every probability space isomorphic to R (with standard algebra and measure). This includes all density distributions of any dimensions.

    In particular the statement that this entropy is equal to the entropy relatively to Lebesgues measure writtent here as an integral of f(x)*log(f(x)) is false.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2022

    Thanks for the alert. Could you say which section of the entry you are referring to, and/or which equation? – I have now made all equations carry a numbering.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeSep 20th 2022
    • (edited Sep 20th 2022)

    In quantum mechanics, the basic notion is the von Neumann entropy defined in terms of density matrix. For type III von Neumann algebras the density matrix is not well defined (physically, the problem is usually in ultraviolet divergences). Von Neumann entropy is generalized to arbitrary semifinite von Neumann algebra in

    • I. E. Segal, A note on the concept of entropy, J. Math. Mech. 9 (1960) 623–629

    …

    A note relating Irving Segal’s notion to relative entropy is

    (Please check)

    diff, v59, current

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 11th 2023

    Added:

    Category-theoretic characterizations of entropy are discussed in

    diff, v61, current

    • CommentRowNumber8.
    • CommentAuthorperezl.alonso
    • CommentTimeSep 17th 2024

    pointer

    diff, v63, current

    • CommentRowNumber9.
    • CommentAuthorkensmosis
    • CommentTimeOct 7th 2024

    Does anyone happen to know a book or review paper that discusses the notion of entropy for a probability field (i.e. the supremum over finite disjoint covers of XX via elements of the σ\sigma-algebra) that is mentioned here? It’s very interesting, but I don’t seem to be able to locate any references to this particular definition. Also, why is it over finite covers of XX rather than countable ones? A σ\sigma-algebra is closed under countable unions, so this puzzles me. I’m sure there must be a good reason, perhaps related to the supremum as a limit, but it’s not immediately obvious to me.

1 to 9 of 9

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