MUBs inequivalence and affine planes

TL;DR

This paper explores the deep connection between large, inequivalent sets of mutually unbiased bases in complex vector spaces and finite affine planes, highlighting their mathematical relationship and summarizing lesser-known results.

Contribution

It clarifies the relationship between inequivalent MUBs and affine planes, providing a summary of lesser-known results in this area.

Findings

Large families of MUBs are unitarily inequivalent in C^N.

The number of MUBs sets grows faster than any polynomial in N.

There is a significant relationship between MUBs and affine planes.

Abstract

There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes "old" results that do not appear to be well-known concerning known families of complete sets of MUBs and their associated planes.