On the connection between mutually unbiased bases and orthogonal Latin squares

TL;DR

This paper investigates the relationship between mutually unbiased bases and orthogonal Latin squares, revealing that their connection may not be as strong in certain composite dimensions as previously thought.

Contribution

It demonstrates that the previously observed correspondence between MUBs and MOLS in prime power dimensions does not extend to some composite dimensions, challenging existing assumptions.

Findings

Complete sets of MUBs are generated in prime power dimensions.

In dimension six, only three MUBs are generated.

The correspondence between MUBs and MOLS breaks down in certain composite dimensions.

Abstract

We offer a piece of evidence that the problems of finding the number of mutually unbiased bases (MUB) and mutually orthogonal Latin squares (MOLS) might not be equivalent. We study a particular procedure which has been shown to relate the two problems and generates complete sets of MUBs in power-of-prime dimensions and three MUBs in dimension six. For these cases, every square from an augmented set of MOLS has a corresponding MUB. We show that this no longer holds for certain composite dimensions.