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Equiangular Vectors Approach to Mutually Unbiased Bases
Faculté des Sciences et Technologies, Université de Lyon, 37 rue du repos, 69361 Lyon, France
Département de Physique, Université Claude Bernard Lyon 1, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France
Groupe Théorie, Institut de Physique Nucléaire, CNRS/IN2P3, 4 rue Enrico Fermi, 69622 Villeurbanne, France
Received: 26 April 2013; Accepted: 6 May 2013 / Published: 8 May 2013
Abstract: 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.
Keywords: finite-dimensional quantum mechanics; mutually unbiased bases; projection operators; positive-semidefinite Hermitian operators; equiangular lines; Gauss sums
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MDPI and ACS Style
Kibler, M.R. Equiangular Vectors Approach to Mutually Unbiased Bases. Entropy 2013, 15, 1726-1737.
Kibler MR. Equiangular Vectors Approach to Mutually Unbiased Bases. Entropy. 2013; 15(5):1726-1737.
Kibler, Maurice R. 2013. "Equiangular Vectors Approach to Mutually Unbiased Bases." Entropy 15, no. 5: 1726-1737.