Equiangular vectors approach to mutually unbiased bases (Q742720)

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