A gap for the maximum number of mutually unbiased bases

A collection of (pairwise) mutually unbiased bases (in short: MUB) in d > 1 dimensions may consist of at most d + 1 bases. Such “complete” collections are known to exists in C when d is a power of a prime. However, in general little is known about the maximum number N(d) of bases that a collection of MUB in C can have. In this work it is proved that given a collection of d MUB in C can be always completed. Hence N(d) 6= d and when d > 1 we have a dichotomy: either N(d) = d + 1 (so that there exists a complete collection of MUB) or N(d) ≤ d−1. In the course of the proof an interesting new characterization is given for a linear subspace of Md(C) to be a subalgebra.

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