Discrete phase-space approach to mutually orthogonal Latin squares

TL;DR

This paper explores the connection between Latin squares, mutually unbiased bases, and phase-space structures in finite dimensions, revealing how unitary equivalences relate to Latin square isomorphisms.

Contribution

It introduces a geometric phase-space framework linking Latin squares and MUBs, and proposes a conjecture relating unitary equivalence of MUBs to Latin square isomorphism.

Findings

Established a correspondence between Latin squares and phase-space monomials.

Derived conditions under which Latin squares are isomorphic based on MUB equivalence.

Proposed a conjecture connecting MUB unitarity to Latin square isomorphism.

Abstract

We show there is a natural connection between Latin squares and commutative sets of monomials defining geometric structures in finite phase-space of prime power dimensions. A complete set of such monomials defines a mutually unbiased basis (MUB) and may be associated with a complete set of mutually orthogonal Latin squares (MOLS). We translate some possible operations on the monomial sets into isomorphisms of Latin squares, and find a general form of permutations that map between Latin squares corresponding to unitarily equivalent mutually unbiased sets. We extend this result to a conjecture: MOLS associated to unitarily equivalent MUBs will always be isomorphic, and MOLS associated to unitarily inequivalent MUBs will be non-isomorphic.