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Equiangular Vectors Approach to Mutually Unbiased Bases

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.
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M. Kibler. Equiangular Vectors Approach to Mutually Unbiased Bases. Entropy, MDPI, 2013, 15, pp.1726-1737. ⟨10.3390/e15051726⟩. ⟨in2p3-00837828⟩

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