Separability criteria for genuine multiparticle entanglement

We present a method to derive separability criteria for different classes of multiparticle entanglement, especially genuine multiparticle entanglement. The resulting criteria are necessary and sufficient for certain families of states. This, for example, completely solves the problem of classifying N-qubit Greenberger-Horne-Zeilinger states mixed with white noise according to their separability and entanglement properties. Further, the criteria are superior to all known entanglement criteria for many other families; also they allow the detection of bound entanglement. We next demonstrate that they are easily implementable in experiments and discuss applications to the decoherence of multiparticle entangled states.