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Entropy 2013, 15(5), 1726-1737; doi:10.3390/e15051726

Equiangular Vectors Approach to Mutually Unbiased Bases

1
Faculté des Sciences et Technologies, Université de Lyon, 37 rue du repos, 69361 Lyon, France
2
Département de Physique, Université Claude Bernard Lyon 1, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France
3
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
(This article belongs to the Special Issue Quantum Information 2012)
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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 finite-dimensional quantum mechanics; mutually unbiased bases; projection operators; positive-semidefinite Hermitian operators; equiangular lines; Gauss sums
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Kibler, M.R. Equiangular Vectors Approach to Mutually Unbiased Bases. Entropy 2013, 15, 1726-1737.

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