论文标题
明确的统一设计的构造
Explicit construction of exact unitary designs
论文作者
论文摘要
本文的目的是为所有$ t $和$ d $提供明确的$ t $ designs $ t $ designs。迄今为止，显式结构似乎仅在非常特殊的情况下闻名。在这里，明确的构造意味着单位矩阵的条目由某些给定多项式的根部函数的值给出。我们将讨论这些方法获得的$ u（4）$中最好的$ 4 $ designs。 确实，我们使用Gelfand Pairs $（g，k）$对紧凑型组的设计进行了感应构造。请注意，$（u（n），u（m）\ times u（n-m））$是gelfand对。通过使用$（g，k）$的区域球形函数，我们可以从$ k $上的$ g $上构建设计。 我们指出的是，我们的证明使用紧凑型群体的表示理论。我们还指出，该方法可以应用于正交组$ o（d）$，因此可以通过$ d $的归纳来在$ d $ dimentional sphere $ s^{d-1} $上对球形$ t $ designs的另一个明确构造。
The purpose of this paper is to give explicit constructions of unitary $t$-designs in the unitary group $U(d)$ for all $t$ and $d$. It seems that the explicit constructions were so far known only for very special cases. Here explicit construction means that the entries of the unitary matrices are given by the values of elementary functions at the root of some given polynomials. We will discuss what are the best such unitary $4$-designs in $U(4)$ obtained by these methods. Indeed we give an inductive construction of designs on compact groups by using Gelfand pairs $(G,K)$. Note that $(U(n),U(m) \times U(n-m))$ is a Gelfand pair. By using the zonal spherical functions for $(G,K)$, we can construct designs on $G$ from designs on $K$. We remark that our proofs use the representation theory of compact groups crucially. We also remark that this method can be applied to the orthogonal groups $O(d)$, and thus provides another explicit construction of spherical $t$-designs on the $d$ dimensional sphere $S^{d-1}$ by the induction on $d$.
