Mutually Unbiased Bases and Orthogonal Latin Squares -- version 3

TL;DR

This paper establishes a link between mutually unbiased bases and Latin squares, proving that a complete set of MUBs exists only in prime power dimensions and confirming their non-existence in dimension six.

Contribution

It demonstrates that the existence of complete MUBs implies the existence of complete MOLSs, and proves the non-existence of complete MUBs in dimension six.

Findings

Complete MUBs imply complete MOLSs.

No complete MUBs in dimension six.

MUBs exist only in prime power dimensions.

Abstract

In this paper, we prove that the existence of a complete set of mutually unbiased bases (MUBs) in N-dimensional Hilbert space implies the existence of a complete set of mutually orthogonal Latin squares (MOLSs) of order N. In particular, we prove that a complete set of MUBs does not exist in dimension six (the first dimension which is not a power of prime).