An infinite family of strongly unextendible mutually unbiased bases in ${\mathbb{C}}^{2^{2h}}$

Abstract

A set of $b$ mutually unbiased bases (MUBs) in ${\mathbb{C}}^d$ (for $d>1$) comprises $bd$ vectors in ${\mathbb{C}}^d$, partitioned into $b$ orthogonal bases for ${\mathbb{C}}^d$ such that the pairwise angle between all vectors from distinct bases is $\arccos (1/\sqrt{d})$. The largest number $\mu (d)$ of MUBs that can exist in ${\mathbb{C}}^d$ is at most $d+1$, but constructions attaining this bound are known only when $d$ is a prime power. A set of $b$ MUBs in ${\mathbb{C}}^d$ that cannot be enlarged, even by the first vector of a potential $(b+1)$-th MUB, is called strongly unextendible. Until now, only one infinite family of dimensions $d$ containing $b(d)$ strongly unextendible MUBs in ${\mathbb{C}}^d$ satisfying $b(d)<\mu (d)$ was known, this family, due to Szántó, is asymptotically "large" in the sense that $b(d)/\mu (d)\to 1$ as $d\to \infty$. However, the existence of $2^{m-1}+1$ strongly unextendible MUBs in ${\mathbb{C}}^{2^m}$ for each integer $m>1$ has been conjectured by Mandayam et al. We prove their conjecture for all even values of $m$, using only elementary linear algebra. The existence of this "small" new infinite family suggests, contrary to widespread belief, that $\mu (d)$ for non-prime-powers $d$ might be significantly larger than the size of particular unextendible sets.

9 pages