A gap for the maximum number of mutually unbiased bases

Weiner, Mihály

doi:10.1090/S0002-9939-2013-11487-5

A gap for the maximum number of mutually unbiased bases
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Proc. Amer. Math. Soc. 141 (2013), 1963-1969 Request permission

Abstract:

A collection of pairwise mutually unbiased bases (in short: MUB) in $d>1$ dimensions may consist of at most $d+1$ bases. Such “complete” collections are known to exist in $\mathbb {C}^d$ when $d$ is a power of a prime. However, in general, little is known about the maximum number $N(d)$ of bases that a collection of MUB in $\mathbb {C}^d$ can have.

In this work it is proved that a collection of $d$ MUB in $\mathbb {C}^d$ can always be completed. Hence $N(d)\neq d$, and when $d>1$ we have a dichotomy: either $N(d)=d+1$ (so that there exists a complete collection of MUB) or $N(d)\leq d-1$. In the course of the proof an interesting new characterization is given for a linear subspace of $M_d(\mathbb {C})$ to be a subalgebra.

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