Affine Constellations Without Mutually Unbiased Counterparts

TL;DR

This paper investigates the relationship between affine constellations and mutually unbiased bases, revealing discrepancies that challenge the conjectured equivalence in certain dimensions, especially six.

Contribution

It introduces affine constellations and compares their existence with mutually unbiased constellations, providing evidence against their equivalence in some cases.

Findings

Discrepancies between affine and mutually unbiased constellations in dimension six

Evidence that the conjectured equivalence may not hold universally

Insights into the distinct nature of these combinatorial structures

Abstract

It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations, mostly in dimension six. The observed discrepancies make a deeper relation between the two existence problems unlikely.