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 Sz\'ant\'o, 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.