MUBs, Polytopes, and Finite Geometries

Abstract

A complete set of N + 1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = pk, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = pk. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.