Two orthonormal bases in dimension d are said “mutually unbiased” if the hermitian scalar products of any vector in one base with any vector in the other base are in modulus equal to the inverse square root of d. One important issue is to determine the maximum number of mutually unbiased bases (MUB) for d dimensions. If d is a prime number or a power of a prime number it is known that this maximum number is d+1. We revisit this problem in both cases using the simple notions of Discrete Fourier Transform, of unitary Hadamard matrices, and of circulant matrices (for which the successive rows are circulant permutations of each other)
