Similarities between mutually unbiased bases and projective planes

Joanne Hall

Abstract

Mutually unbiased bases (MUBs) are a structure of importance in quantum information theory. It is known that there can be a maximum of d+1 in a d dimensional complex vector space, such sets of d+1 MUBs are called complete. It has been conjectured that the existence of a complete set of MUBs is linked to the existence of a projective plane of order d. I discuss some of the evidence for and against this conjecture.