Equiangular Vectors Approach to Mutually Unbiased Bases

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 C-d, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space C-d (with a modulus for the inner product) into the one of finding d (d + 1) vectors in the d(2)-dimensional space C-d2 (without a modulus for the inner product). The transformation from C-d to C-d2 corresponds to the passage from equiangular lines to equiangular vectors. The transformation formulas are discussed in the case where d is a prime number.