论文标题
来自对称组的正图和痕量多项式
Positive maps and trace polynomials from the symmetric group
论文作者
论文摘要
借助量子信息理论借来的技术，我们开发了一种方法来系统地获得多个矩阵变量中的操作员不等式和身份。这些以痕量多项式的形式：类似多项式的表达式，涉及矩阵单元素$ x_ {α_1} \ cdots x_ {α_r} $及其痕迹$ \ operatotorName {tr}（x_ {x_ {x_1}} x__________________ {α_r}α_r} $。我们的方法在于将对称组对张量产物空间的作用转化为矩阵乘法的作用。结果，我们将两极分化的Cayley-Hamilton身份扩展到正锥上的操作员不平等，以Werner状态证人的身份表征了一组多线性均衡的正地图，并在张量产品空间上构造了置换多项式多项式和张量的多项式标识。我们与量子信息理论和不变理论中的概念有联系。
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials $X_{α_1} \cdots X_{α_r}$ and their traces $\operatorname{tr}(X_{α_1} \cdots X_{α_r})$. Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.
