A rigidity property of complete systems of mutually unbiased bases

Suppose that for some unit vectors b1, . . .bn in C d we have that for any j 6= k bj is either orthogonal to bk or |〈bj ,bk〉| = 1/d (i.e. bj and bk 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.

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