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Henri Poincar\u00e9"],"published-print":{"date-parts":[[2023,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We address a recent conjecture stated by Z. Van Herstraeten and N. J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the R\u00e9nyi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb\u2019s inequalities.\n<\/jats:p>","DOI":"10.1007\/s00023-023-01298-x","type":"journal-article","created":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T16:02:46Z","timestamp":1681228966000},"page":"2341-2362","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["On a Recent Conjecture by Z. Van Herstraeten and N. J. Cerf for the Quantum Wigner Entropy"],"prefix":"10.1007","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9019-8406","authenticated-orcid":false,"given":"Nuno Costa","family":"Dias","sequence":"first","affiliation":[]},{"given":"Jo\u00e3o Nuno","family":"Prata","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,11]]},"reference":[{"key":"1298_CR1","unstructured":"Babenko, K.I.: An inequality in the theory of Fourier integrals. Izv. Akad. Nauk SSSR Ser. Mat. 25,: 531\u2013542. English trans. Amer. Math. Soc. Transl. 44, 115\u2013128 (1961)"},{"key":"1298_CR2","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/ac55eb","volume":"55","author":"P Balazs","year":"2022","unstructured":"Balazs, P., Teofanov, N.: Continuous frames in tensor product Hilbert spaces, localization operators and density operators. J. Phys. A: Math. Theor. 55, 145201 (2022)","journal-title":"J. Phys. A: Math. 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