$\mathbb{C}^{2^{2h}}$中强不可扩互无偏基的无限族
An infinite family of strongly unextendible mutually unbiased bases in $\mathbb{C}^{2^{2h}}$
摘要：
$\mathbb{C}^d$（对于$d > 1$）中的一组$b$相互无偏碱基（MUB）包括$\mathbb{C}^d$中的$bd$载体，其被划分成$\mathbb{C}^d$的$b$正交碱基，使得来自不同碱基的所有载体之间的成对角度为$\arccos(1/\sqrt{d})$。在$\mathbb{C}^d$中可以存在的MUB的最大数目$μ(d)$至多是$d+1$，但是只有当$d$是素数幂时，才知道达到这个界限的构造。$\mathbb{C}^d$中的一组$b$ MUB即使不能被潜在的第$(b+1)$个MUB的第一个向量扩大，也被称为强不可扩展的。到目前为止，只有一个无限的家庭的尺寸$d$包含$b(d)$强烈不可扩展MUB在$\mathbb{C}^d$满足$b(d) < μ(d)$是已知的，这个家庭，由于Szántó，是渐近“大”的意义上说，$b(d)/μ(d) \to 1$作为$d \to \infty$。然而，存在的$2^{m-1}+1$强不可扩展MUBs在$\mathbb{C}^{2^m}$为每个整数$m > 1$已被证明Mandayam等人。我们证明了他们的猜想，所有甚至值的$m$，仅使用初等线性代数。这个“小”的新无限族的存在表明，与普遍的看法相反，非素数幂PDMATHAMT的PDMATHAMR可能比特定的不可扩展集的大小要大得多。
英文摘要：
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 $μ(d)$ 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 $b(d) < μ(d)$ was known, this family, due to Szántó, is asymptotically "large" in the sense that $b(d)/μ(d) \to 1$ 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 $μ(d)$ for non-prime-powers $d$ might be significantly larger than the size of particular unextendible sets.
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