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In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old conjecture, known as Zauner's conjecture, stating that there exist at most three. Here we tackle Zauner's conjecture numerically through the construction of Bell inequalities for every pair of integers <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>d<\/mml:mi><mml:mo>&amp;#x2265;<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math> that can be maximally violated in dimension <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math> if and only if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math> MUBs exist in that dimension. Hence we turn Zauner's conjecture into an optimisation problem, which we address by means of three numerical methods: see-saw optimisation, non-linear semidefinite programming and Monte Carlo techniques. All three methods correctly identify the known cases in low dimensions and all suggest that there do not exist four mutually unbiased bases in dimension six, with all finding the same bases that numerically optimise the corresponding Bell inequality. Moreover, these numerical optimisers appear to coincide with the ``four most distant bases'' in dimension six, found through numerically optimising a distance measure in [P. Raynal, X. L\u00fc, B.-G. Englert, {Phys. Rev. A}, { 83} 062303 (2011)]. Finally, the Monte Carlo results suggest that at most three MUBs exist in dimension ten.<\/jats:p>","DOI":"10.22331\/q-2022-08-17-778","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T08:51:11Z","timestamp":1660726271000},"page":"778","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":["Three numerical approaches to find mutually unbiased bases using Bell inequalities"],"prefix":"10.22331","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7866-8356","authenticated-orcid":false,"given":"Maria Prat","family":"Colomer","sequence":"first","affiliation":[{"name":"ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain"},{"name":"CFIS-Centre de Formaci\u00f3 Interdisciplin\u00e0ria Superior, UPC-Universitat Polit\u00e8cnica de Catalunya, 08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5644-8985","authenticated-orcid":false,"given":"Luke","family":"Mortimer","sequence":"additional","affiliation":[{"name":"ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7703-8539","authenticated-orcid":false,"given":"Ir\u00e9n\u00e9e","family":"Fr\u00e9rot","sequence":"additional","affiliation":[{"name":"ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain"},{"name":"Univ Grenoble Alpes, CNRS, Grenoble INP, Institut N\u00e9el, 38000 Grenoble, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2682-8215","authenticated-orcid":false,"given":"M\u00e1t\u00e9","family":"Farkas","sequence":"additional","affiliation":[{"name":"ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1355-3435","authenticated-orcid":false,"given":"Antonio","family":"Ac\u00edn","sequence":"additional","affiliation":[{"name":"ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain"},{"name":"ICREA-Institucio Catalana de Recerca i Estudis Avan\u00e7ats, Lluis Companys 23, 08010 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2022,8,17]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"I D Ivanovic. 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