{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:23:24Z","timestamp":1772252604413,"version":"3.50.1"},"reference-count":7,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,2,29]],"date-time":"2020-02-29T00:00:00Z","timestamp":1582934400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The famous singlet correlations of a composite quantum system consisting of two two-level components in the singlet state exhibit notable features of two kinds. One kind are striking certainty relations: perfect anti-correlation, and perfect correlation, under certain joint settings. The other kind are a number of symmetries, namely invariance under a common rotation of the settings, invariance under exchange of components, and invariance under exchange of both measurement outcomes. One might like to restrict attention to rotations in the plane since those are the ones most commonly investigated experimentally. One can then also further distinguish between the case of discrete rotations (e.g., only settings which are a whole number of degrees are allowed) and continuous rotations. We study the class of classical correlation functions, i.e., generated by classical physical systems, satisfying all these symmetries, in the continuous, planar, case. We call such correlation functions classical EPR-B correlations. It turns out that if the certainty relations and rotational symmetry holds at the level of the correlations, then rotational symmetry can be imposed \u201cfor free\u201d on the underlying classical physical model by adding an extra randomisation level. The other binary symmetries are obtained \u201cfor free\u201d. This leads to a simple heuristic description of all possible classical EPR-B correlations in terms of a \u201cspinning bi-coloured disk\u201d model. We deliberately use the word \u201cheuristic\u201d because technical mathematical problems remain wide open concerning the transition from finite or discrete to continuous. The main purpose of this paper is to bring this situation to the attention of the mathematical community. We do show that the widespread idea that \u201cquantum correlations are more extreme than classical physics would allow\u201d is at best highly inaccurate, through giving a concrete example of a classical correlation which satisfies all the symmetries and all the certainty relations and which exceeds the quantum correlations over a whole range of settings. It is found by a search procedure in which we randomly generate classical physical models and, for each generated model, evaluate its properties in a further Monte-Carlo simulation of the model itself.<\/jats:p>","DOI":"10.3390\/e22030287","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T04:16:16Z","timestamp":1583122576000},"page":"287","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Triangle Wave Versus the Cosine: How Classical Systems Can Optimally Approximate EPR-B Correlations"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5821-9986","authenticated-orcid":false,"given":"Richard David","family":"Gill","sequence":"first","affiliation":[{"name":"Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195","article-title":"On the Einstein Podolsky Rosen paradox","volume":"1","author":"Bell","year":"1964","journal-title":"Physics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1103\/PhysRev.47.777","article-title":"Can quantum-mechanical description of physical reality be considered complete?","volume":"16","author":"Einstein","year":"1935","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1103\/PhysRev.108.1070","article-title":"Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky","volume":"108","author":"Bohm","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"062124","DOI":"10.1103\/PhysRevA.90.062124","article-title":"Bloch-sphere colorings and Bell inequalities","volume":"90","author":"Kent","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_5","unstructured":"Wang, F. (2014). Bell tests with optimal local hidden variable models. arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"187904","DOI":"10.1103\/PhysRevLett.91.187904","article-title":"Communication Cost of Simulating Bell Correlations","volume":"91","author":"Toner","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","unstructured":"Palmer, T.N. (2019). Bell Inequality Violation with Free Choice and Local Causality on the Invariant Set. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/287\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:02:45Z","timestamp":1760173365000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/287"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,29]]},"references-count":7,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["e22030287"],"URL":"https:\/\/doi.org\/10.3390\/e22030287","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201912.0381.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,29]]}}}