Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An infinite family of strongly unextendible mutually unbiased bases in
  $\mathbb{C}^{2^{2h}}$   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.

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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci