Detection of Multiparticle Entanglement: Quantifying the Search for Symmetric Extensions

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We provide quantitative bounds on the characterisation 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 LOCC or Frobenius norm). Our results provide a theoretical justification for the use of the Search for Symmetric Extensions as a practical test for multiparticle entanglement.