A Rigidity Property of Complete Systems of Mutually Unbiased Bases

Suppose that for some unit vectors b(1), ... b(n) in C-d we have that for any j not equal k b(j) is either orthogonal to b(k) or vertical bar < b(j), b(k)>vertical bar(2) = 1/d (i.e., b(j) and b(k) are unbiased). We prove that if n = d(d + 1), then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into d + 1 orthonormal bases, all being mutually unbiased with respect to each other.