具有循环块的块循环矩阵，良好和互无偏基，II。原动力案例
Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
摘要：
在我们以前的文件\cite{co1}中，我们已经表明，循环矩阵的理论允许恢复的结果，存在$p+1$相互无偏基的尺寸$p$，$p$是一个任意的素数。如果$\forall b\in \mathcal B, \forall b' \in \mathcal B'$之一具有该$$| b\cdot b'| = \frac{1}{\sqrt d}$$（$\mathbb C^d$中的$b\cdot b'$厄米标量积），则称$\mathbb C^d$的两个正交基$\mathcal B, \mathcal B'$相互无偏。本文证明了具有循环块的块循环矩阵的理论可以非常简单地证明已知的结果：如果$d=p^n$（$p$是素数，$n$是任意整数），则在$\mathbb C^d$中存在$d+1$互无偏基.我们的结果在很大程度上依赖于Klimov，Munoz，Romero \cite{klimuro}的一个想法。作为一个子产品，我们恢复二次韦伊和$p\ge 3$的性质，这推广了这样一个事实，即在素数的情况下，二次高斯和性质遵循我们的结果。
英文摘要：
In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases $\mathcal B, \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if $\forall b\in \mathcal B, \forall b' \in \mathcal B'$ one has that $$| b\cdot b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbb C^d$). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if $d=p^n$ ($p$ a prime number, $n$ any integer) there exists $d+1$ mutually Unbiased Bases in $\mathbb C^d$. Our result relies heavily on an idea of Klimov, Munoz, Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil sums for $p\ge 3$, which generalizes the fact that in the prime case the quadratic Gauss sums properties follow from our results.
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