Talk:Von Neumann entropy

Concavity
I think the definition of concavity should include a requirement that $$\sum_i \lambda_i = 1$$, but I'm not 100% sure. See, for example Alfred Wehrl, Reviews of Modern Physics,50,2,1978. Njerseyguy (talk) 05:55, 10 July 2009 (UTC)


 * Of, of course. Just choose a single non-zero $$\lambda > 1$$.  Then it's easy to see that the formula doesn't hold.  I've put in the condition. Njerseyguy (talk) 07:30, 10 July 2009 (UTC)

Base of logarithm
The two references I found use ln as the definition, so I made the article consistent with that. However, there is some indication that some authors use base-2 logarithm instead. Someone should confirm that in a reference and then the article can state that both alternatives are in use. --Steve (talk) 20:22, 28 April 2012 (UTC)

On https://en.wikipedia.org/wiki/Quantum_entanglement the log2 version is written. I feel like the ln one is the most common. — Preceding unsigned comment added by 132.229.215.11 (talk) 12:40, 5 March 2018 (UTC)

Extension to Shannon?
Von Neumann predates Shannon. Shouldn't we say Shannon entropy is an extension of von Neumann entropy to information theory? -- D昌양 ( Talk ) 21:02, 31 August 2015 (UTC)

Might be helpful (short answer is yes): Foreword of the Proceedings 'The Maximum Entropy Formalism' (1978, isbn 0-262-12080-1): "[...] Some 100 years ago Boltzmann (1) presented the notion of the most probable distribution, the forerunner of the maximum entropy procedure. About 75 years ago, Planck (2) maximizes the entropy so as to derive the distribution which bears his name and which changed the course of physics, and Gibbs (3) presented a concise and preciser statement that is still a source of inspiration. Some 50 years ago von Neumann (4) introduced entropy into quantum mechanics and Hartley (5) sought to define the concept of information. Forty years ago Elsasser (6) suggested that the entropy of a quantal system should be maximized. Thirty years ago, Cox (7) discussed the algebra of probable inference, while Shannon (8) and Wiener (9) sought to refine the concept of information, with Winer putting particular stress on physical applications. Finally, 20 years ago, Jaynes (10) invoke the Shannon axiomatic characterization of the 'uncertainty' of entropy function and the Cox interpretation of probability as reflecting a state of knowledge, to cast the principle in its general form, as a principle of inductive reasoning independent of any specific application. The more recent history will be found in the pages of this volume which also includes the contributions by Prof. Elsasser, Cox, and Jaynes. [...]" -- M. Tribus, R. D. Levine 1. Boltzmann 1877 "über die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective, den Satz über das Wärmegleichgewicht" Wein Ber. 76:373-95 2. Plack 1906 "Theory der Wärmestrahlung" Leipzig: J. A. Barth 3. Gibbs, J. W. 1902 "Elementary Principles in Statistical Mechanics" New Haven: Yale University 4. von Neumann 1927 "Thermodynamik quantenmechanischer Gesamtheiten" Gott. Nach. 273-91 5. Hartley R.V.L. 1928 "Transmission of Information" Bell Syst. Tech. J. 7:535-63 6. Elsasser, W. M. 1937 "On Quantum Measurements and the Role of Unvertainty Relations in Statistical Mechnaics" Phys. Rev. 52:987-999 7. Cox R.T. 1946, "Probability, Freqency and Reasoinable Expectations" Am. J. Phys 14:1-14 8. Shannon C. E. 1948 "A Mathematical Theory of Communication" Bell Syst. Tech. J. 27:379-423 9. Wiener N. 1948 "Cybernetcis" Cambridge MIT Press 10. Jaynes. E. T. 1956 "Information Theory and Statistical Mechnics" Physr. Rev. 106:620-30 --Ruiin (talk) 15:18, 21 March 2019 (UTC)

Looked at article, for UG student use
This article introduction is not approachable. Then, when getting into the article, it seems to be much original WP editor interpretation. According to MOS:Intro, article introductions are to:

"...avoid difficult-to-understand terminology and symbols. Mathematical equations and formulas should be avoided when they conflict with the goal of making the lead section accessible to as broad an audience as possible."

According to WP:PSTS, sourcing for articles is supposed to comply with the following:

"...sources are needed to establish the topic's notability and to avoid novel interpretations of primary sources. All analyses and interpretive or synthetic claims about primary sources must be referenced to a secondary or tertiary source, and must not be an original analysis of the primary-source material by Wikipedia editors."

Despite these guidelines, the introduction to this article is a technical presentation focused on equations, and largely jargon only understandable to specialists. And the Background section presents its material citing only two sources (those where key ideas were first developed, including by the section's eponymous author), rather than true secondary/tertiary sources where the original works are interpreted and explained. None of the material can be described as common knowledge. (For instance, the closing unreferenced sentence of the Background section states, "Mathematically, $$\hat{\rho}$$ is a positive-semidefinite Hermitian matrix with unit trace.")

Without these further explanatory sources, it appears that the editors here are reading the original works, and interpreting and explaining them to us, themselves. We understand this to be against Wikipedia policy.

For these reasons, the article cannot recommended to students; it lacks sourcing relevant to the interpretation presented, to allow followup with these sources, for understanding and verification.

While its design is different from Wikipedia, asking for trust of the stated academic author of its articles, the Scholarpedia article on "Quantum Entropies" (by Prof. Anna Vershynina of Univ. of Houston) my be another place to begin looking for suitable introductory student content on the von Neumann subject.

Upper bound of the entropy
I believe the upper bound for the entropy to be incorrect, it should be in my opinion ln(N) instead of N/e. I have confirmed in another website that it is indeed the case and so I will proceed to change it. I will redact it if I am mistaken. 2001:8A0:FF83:400:4C4D:5949:922C:78B2 (talk) 20:49, 18 January 2023 (UTC)

Strong subadditivity of quantum entropy
The sentence about Kiefer does not seem to accurately reflect what's in the references [9,10] and what's written under the main Wikipedia article about Strong subadditivity of quantum entropy:

"J. Kiefer proved a peripherally related convexity result in 1959, which is a corollary of an operator Schwarz inequality proved by E.H.Lieb and M.B.Ruskai. However, these results are comparatively simple, and the proofs do not use the results of Lieb's 1973 paper on convex and concave trace functionals. It was this paper that provided the mathematical basis of the proof of SSA by Lieb and Ruskai. The extension from a Hilbert space setting to a von Neumann algebra setting, where states are not given by density matrices, was done by Narnhofer and Thirring." Request for correction (talk) 15:28, 14 February 2023 (UTC)