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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 7th 2021

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

• 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

1. Fixed a broken link in the references.

Rongmin Lu

2. Added details to a reference.

Rongmin Lu

• 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
• CommentTime5 days ago
• (edited 5 days ago)

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