A gap for the maximum number of mutually unbiased bases

Abstract

A collection of pairwise mutually unbiased bases (in short: MUB) in d > 1 d>1 dimensions may consist of at most d + 1 d+1 bases. Such “complete” collections are known to exist in C d \mathbb {C}^d when d d is a power of a prime. However, in general, little is known about the maximum number N ( d ) N(d) of bases that a collection of MUB in C d \mathbb {C}^d can have. In this work it is proved that a collection of d d MUB in C d \mathbb {C}^d can always be completed. Hence N ( d ) ≠ d N(d)eq d , and when d > 1 d>1 we have a dichotomy: either N ( d ) = d + 1 N(d)=d+1 (so that there exists a complete collection of MUB) or N ( d ) ≤ d − 1 N(d)\leq d-1 . In the course of the proof an interesting new characterization is given for a linear subspace of M d ( C ) M_d(\mathbb {C}) to be a subalgebra.