更多互不偏基

中文摘要 AI 辅助

互不偏基（MUBs）描述了这样一种量子测量：在一个基中确定的结果在其他基中给出均匀的结果。在不是素数幂的维度中，它们的最大数量仍然未知。我们引入了一个ansatz（试探性构造），在许多维度中，它使我们能够构造出比以前发现的更多的MUBs。该ansatz中的每个基由一个对角相位矩阵作用于一个固定的酉矩阵构成，该酉矩阵是傅里叶矩阵的张量积，并且可选地，可以是任何实Hadamard矩阵。这种构造在维度12中产生5个MUBs，在维度48、96和192中产生6个，在维度36和108中产生7个，在维度648中产生10个。这些基随后可以扩展，以渐近地超过张量积界限，在每个第十八维度中。此外，我们证明了Paley的实Hadamard矩阵可用于在维度$d = q(q+1)$中构造$q+1$个基，其中$q$是满足$q\equiv3\pmod4$的素数幂。这个数量随$\sqrt d$增长，并且不能被较小集合的张量积所超过。

英文摘要

Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct $q+1$ bases in dimension $d = q(q+1)$ for every prime power $q\equiv3\pmod4$. This count grows as $\sqrt d$ and cannot be exceeded by tensor products of smaller sets.

发表机构

• Yale University（耶鲁大学）
• The Flatiron Institute（平顿研究所）

机构由 AI 辅助整理，请以论文原文为准。