Weyl幺正的普适性
Universality of Weyl Unitaries
摘要：
魏尔的酉矩阵，这是介绍了魏尔的1927年论文群论和量子力学，是$p\times p$酉矩阵所给出的对角矩阵的条目是$p\times p$-th根单位和循环移位矩阵。Weyl幺正，我们用$\mathfrak u$和$\mathfrak v$表示，满足$\mathfrak u^p=\mathfrak v^p=1_p$（$p\times p$单位矩阵）和对易关系$\mathfrak u\mathfrak v=ζ\mathfrak v\mathfrak u$，其中$ζ$是$p$的本原单位根.本文证明了Weyl酉矩阵在以下意义下是泛矩阵：若$u$和$v$是任意的$d\times d$酉矩阵，使得$u^p= v^p=1_d$和$ u v=ζvu$，则存在有单位的完全正线性映射$ϕ:\mathcal M_p(\mathbb C)\rightarrow\mathcal M_d(\mathbb C)$，使得$ϕ(\mathfrak u)= u$和$ϕ(\mathfrak v)=v$ .此外，我们还表明，任何两对PDMATHAMN阶酉矩阵，满足Weyl交换关系是完全为了等价的。当$p=2$时，外尔矩阵是量子力学中三个泡利矩阵中的两个。最近表明，$g$元组的Pauli-Weyl-Brauer酉是通用的所有$g$元组的反交换自伴酉矩阵，但是，我们在这里表明，类似的结果失败的正整数$p>2$。最后，我们证明了Weyl矩阵是极值在其矩阵范围内，使用最近的想法从非交换凸理论。
英文摘要：
Weyl's unitary matrices, which were introduced in Weyl's 1927 paper on group theory and quantum mechanics, are $p\times p$ unitary matrices given by the diagonal matrix whose entries are the $p$-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by $\mathfrak u$ and $\mathfrak v$, satisfy $\mathfrak u^p=\mathfrak v^p=1_p$ (the $p\times p$ identity matrix) and the commutation relation $\mathfrak u\mathfrak v=ζ\mathfrak v\mathfrak u$, where $ζ$ is a primitive $p$-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if $u$ and $v$ are any $d\times d$ unitary matrices such that $u^p= v^p=1_d$ and $ u v=ζvu$, then there exists a unital completely positive linear map $ϕ:\mathcal M_p(\mathbb C)\rightarrow\mathcal M_d(\mathbb C)$ such that $ϕ(\mathfrak u)= u$ and $ϕ(\mathfrak v)=v$. We also show, moreover, that any two pairs of $p$-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent. When $p=2$, the Weyl matrices are two of the three Pauli matrices from quantum mechanics. It was recently shown that $g$-tuples of Pauli-Weyl-Brauer unitaries are universal for all $g$-tuples of anticommuting selfadjoint unitary matrices; however, we show here that the analogous result fails for positive integers $p>2$. Finally, we show that the Weyl matrices are extremal in their matrix range, using recent ideas from noncommutative convexity theory.
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