  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.

Quantum Physics