Dimension-free entanglement detection in multipartite Werner states

TL;DR

This paper characterizes entanglement in multipartite Werner states independently of local Hilbert space dimensions, introducing dimension-free witnesses constructed via semidefinite programming hierarchies.

Contribution

It provides a complete, dimension-independent characterization of Werner state entanglement and develops two novel semidefinite programming methods for constructing entanglement witnesses.

Findings

Existence of dimension-free entanglement witnesses for all entangled Werner states

Development of two semidefinite programming hierarchies for witness construction

Convergence of the first hierarchy to a witness for every entangled state

Abstract

Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a complete characterization of their entanglement that is independent of the underlying local Hilbert space: for every entangled Werner state there exists a dimension-free entanglement witness. The construction of such a witness is formulated as an optimization problem. To solve it, two semidefinite programming hierarchies are introduced. The first one is derived using real algebraic geometry applied to positive polynomials in the entries of a Gram matrix, and is complete in the sense that for every entangled Werner state it converges to a witness. The second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables, and is a relaxation that involves smaller semidefinite constraints.