A pair of orthonormal bases in a Hilbert space are said to be mutually unbiased if the square of the inner-product between any pair of vectors from each basis has a constant magnitude, equal to the inverse of the dimension of the space. Mutually unbiased bases (MUBs) lie at the heart of theoretical investigations into complementarity in quantum theory. While much is known about the existence and properties of such bases prime-power dimensions, there are interesting mathematical questions that remain to be answered. In particular, while constructions of maximal sets of (d+1) MUBs in Hilbert spaces of prime-powered dimensions are known, but not a single example of such a set is known in other dimensions (d = 6, 10, etc.). In this talk, we review some of the known constructions of MUBs, touch upon the open problem of whether there exist 7 such bases in a Hilbert space of dimension d=6, and, discuss recent ide as on unextendible sets of MUBs. We will also briefly illustrate the usefulness of such unbiased bases in the context of quantum cryptography and quantum information theory.
