Constructions of Mutually Unbiased Bases

Two orthonormal bases B and B′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |〈b|b′ 〉|2 = 1/d holds for all b ∈ B and b′ ∈ B′. The size of any set containing pairwise mutually unbiased bases of ℂd cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.