Positive maps and trace polynomials from the symmetric group
- URL: http://arxiv.org/abs/2002.12887v2
- Date: Wed, 9 Sep 2020 11:16:00 GMT
- Title: Positive maps and trace polynomials from the symmetric group
- Authors: Felix Huber
- Abstract summary: We develop a method to obtain operator inequalities and identities in several variables.
We give connections to concepts in quantum information theory and invariant theory.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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 $X_{\alpha_1} \cdots X_{\alpha_r}$
and their traces $\operatorname{tr}(X_{\alpha_1} \cdots X_{\alpha_r})$. 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.
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