Mutually Unbiased Bases and the Complementarity Polytope

Abstract

A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N2 - 1 dimensional space of N × 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 = pk, 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 N2 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 = pk; perhaps they do not exist otherwise. However, the link to the existence of MUBs — if any — remains to be found.