Detection of Multiparticle Entanglement: Quantifying the Search for Symmetric Extensions

Abstract

We provide quantitative bounds on the characterization of multiparticle separable states by states that have locally symmetric extensions. The bounds are derived from two-particle bounds and relate to recent studies on quantum versions of de Finetti's theorem. We discuss algorithmic applications of our results, in particular a quasipolynomial-time algorithm to decide whether a multiparticle quantum state is separable or entangled (for constant number of particles and constant error in the norm induced by one-way local operations and classical communication, or in the Frobenius norm). Our results provide a theoretical justification for the use of the search for symmetric extensions as a test for multiparticle entanglement.