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 $X_{{\alpha }_1},\dots ,X_{{\alpha }_r}$ and their traces $\mathrm{tr}(X_{{\alpha }_1},\dots ,X_{{\alpha }_r})$. 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.

中文翻译：

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来自对称组的正图和迹多项式

利用从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取几个矩阵变量中的算子不等式和恒等式。这些采用跟踪多项式的形式：涉及矩阵单项式的类似多项式的表达式$X_{{\alpha }_{\mathrm{1个}}}，\dots ，X_{{\alpha }_{\mathrm{[R}}}$ 及其痕迹 $\mathrm{TR}（X_{{\alpha }_{\mathrm{1个}}}，\dots ，X_{{\alpha }_{\mathrm{[R}}}）$。我们的方法基于将对称组在张量积空间上的作用转换为矩阵相乘的作用。结果，我们将极化的Cayley-Hamilton恒等式扩展到正圆锥上的算子不等式，根据Werner状态见证者表征了多线性等变正映射集，并在张量积空间上构造了置换多项式和张量多项式恒等式。我们将量子信息理论和不变理论中的概念联系起来。