Welch界的一个判据及其在互无偏基上的应用
A Criterion for Attaining the Welch Bounds with Applications for Mutually Unbiased Bases
摘要：
本文简要介绍了互无偏基和Welch界，并证明后者是探索互无偏基的一个很好的技术工具。特别是，一个标准的系统的向量，以满足韦尔奇界限与平等的情况下，MUBs。这产生了一个必要和充分条件的一组标准正交基，以形成一个完整的系统的MUB。这个条件在MUB的齐次系统的情况下采取特别优雅的形式。我们表达了一些已知的MUB结构在这种形式。此外，它示出了最近获得的结果绑定MUBs和一些组合结构（如完美的非线性函数和相对差集）自然遵循这个标准。一些方向证明不存在的结果也勾勒。
英文摘要：
The paper gives a short introduction to mutually unbiased bases and the Welch bounds and demonstrates that the latter is a good technical tool to explore the former. In particular, a criterion for a system of vectors to satisfy the Welch bounds with equality is given and applied for the case of MUBs. This yields a necessary and sufficient condition on a set of orthonormal bases to form a complete system of MUBs. This condition takes an especially elegant form in the case of homogeneous systems of MUBs. We express some known constructions of MUBs in this form. Also it is shown how recently obtained results binding MUBs and some combinatorial structures (such as perfect nonlinear functions and relative difference sets) naturally follow from this criterion. Some directions for proving non-existence results are sketched as well.
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