Dimension-free entanglement detection in multipartite Werner states
Résumé
Werner states are multipartite quantum states that are invariant under the diagonal conjugate action of the unitary group. This paper gives a characterization of their entanglement that is independent of the underlying local Hilbert space. For a given Werner state, the construction of a dimension-free entanglement 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 the second one is based on a sum-of-squares certificate for the positivity of trace polynomials in noncommuting variables.