Positive maps and trace polynomials from the symmetric group

Abstract

This article develops a systematic method to obtain operator inequalities in several matrix variables. These take the form of polynomial-like expressions that involve matrix monomials X α 1 • • • X α r and their traces tr(X α 1 • • • 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, and construct swap polynomials and other matrix identities on tensor product spaces. Further connections to concepts from invariant theory, polynomial identity rings, and multipartite entanglement are given.

Optimization over trace polynomials

“…However, [PKRR + 19] does not provide a proof of convergence for this hierarchy. In [Hub20], the author focuses on the multilinear case and obtains a characterization of all multilinear equivariant trace polynomials which are positive on the positive cone. In a closely related work in real algebraic geometry [K ŠV18], the first and third author derive several Positivstellensätze for trace polynomials positive on semialgebraic sets of fixed size matrices.…”

Positive maps from irreducibly covariant operators

“…Next, in [14] further generalisations are discussed. Namely, authors using elements of the representation theory and graphical representation of permutations, have characterised linear maps Φ : M(n, C) → M(n a , C) ⊗ M(n b , C) satisfying Φ(UXU † ) = ( Ū⊗a ⊗ U ⊗b )Φ(X)( Ū⊗a ⊗ U ⊗b ) † , for all X ∈ M(n, C) and a, b ∈ N. Recently, in paper [15] such covariance, with a = 0, plays the central role in extension of the polarized Cayley-Hamilton identity to an operator inequality on the positive cone and characterisation of the set of multilinear covariant positive maps. Finally, we mention 9 the result from [16], where the family of positive maps induced by the covariance with respect to the finite group generated by the Weyl operators is derived.…”