{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:53Z","timestamp":1760243453216,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2013,5,8]],"date-time":"2013-05-08T00:00:00Z","timestamp":1367971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the other equals 1 d. The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d+1) vectors in the d2-dimensional space Cd2 (without a modulus for the inner product). The transformation from Cd to Cd2 corresponds to the passage from equiangular lines to equiangular vectors. The transformation formulas are discussed in the case where d is a prime number.<\/jats:p>","DOI":"10.3390\/e15051726","type":"journal-article","created":{"date-parts":[[2013,5,8]],"date-time":"2013-05-08T11:26:23Z","timestamp":1368012383000},"page":"1726-1737","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Equiangular Vectors Approach to Mutually Unbiased Bases"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-4561","authenticated-orcid":false,"given":"Maurice","family":"Kibler","sequence":"first","affiliation":[{"name":"Facult\u00e9 des Sciences et Technologies, Universit\u00e9 de Lyon, 37 rue du repos, 69361 Lyon, France"},{"name":"D\u00e9partement de Physique, Universit\u00e9 Claude Bernard Lyon 1, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France"},{"name":"Groupe Th\u00e9orie, Institut de Physique Nucl\u00e9aire, CNRS\/IN2P3, 4 rue Enrico Fermi, 69622 Villeurbanne, France"}]}],"member":"1968","published-online":{"date-parts":[[2013,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1073\/pnas.46.4.570","article-title":"Unitary operator bases","volume":"46","author":"Schwinger","year":"1960","journal-title":"Proc. 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