Mutually unbiased bases and the complementarity polytope

A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N-2 - 1 dimensional space of N x N Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N = p(k), where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N-2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N. Affine planes of order N are known to exist if N = p(k); perhaps they do not exist otherwise. However, the link to the existence of MUBs - if any - remains to be found.