Positive maps and trace polynomials from the symmetric group

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 $\mathrm{tr}(X_{{\alpha }_1}\cdots X_{{\alpha }_r})$. 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.

arXiv comment28 pages, 3 figures, 2 tables. Extensively rewritten: asymmetric maps, proof for Motzkin matrix polynomial, and connections to QIT added. Comments welcome!

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Cited by in corpus (12)

• Analysing quantum systems with randomised measurements
• Optimization over trace polynomials
• Dimension-free entanglement detection in multipartite Werner states
• Entanglement detection with trace polynomials
• Linear programming with unitary-equivariant constraints
• Tensor polynomial identities
• Positive maps from the walled Brauer algebra
• Refuting spectral compatibility of quantum marginals
• Entanglement and distillation from symmetric positive maps
• Experimental measurement and a physical interpretation of quantum shadow enumerators
• Werner states from diagrams
• Positivity of state, trace, and moment polynomials, and applications in quantum information