We present a method to derive separability criteria for the 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.

We derive N-particle Bell-type inequalities under the assumption of partial separability, i.e., that the N-particle system is composed of subsystems which may be correlated in any way (e.g., entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experi-mentally accessible sufficient conditions for full N-particle entanglement, i.e., for N-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the N-particle Greenberger-Horne-Zeilinger states.

We explore the subtle relationships between partial separability and entanglement of subsystems in multiqubit quantum states and give experimentally accessible conditions that distinguish between various classes and levels of partial separability in a hierarchical order. These conditions take the form of bounds on the correlations of locally orthogonal observables. Violations of such inequalities give strong sufficient criteria for various forms of partial inseparability and multiqubit entanglement. The strength of these criteria is illustrated by showing that they are stronger than several other well-known entanglement criteria (the fidelity criterion, violation of Mermin-type separability inequalities, the Laskowski-Żukowski criterion and the Dür-Cirac criterion), and also by showing their great noise robustness for a variety of multiqubit states, including N -qubit GHZ states and Dicke states. Furthermore, for N ≥ 3 they can detect bound entangled states. For all these states, the required number of measurement settings for implementation of the entanglement criteria is shown to be only N + 1. If one chooses the familiar Pauli matrices as single-qubit observables, the inequalities take the form of bounds on the anti-diagonal matrix elements of a state in terms of its diagonal matrix elements.

We point out a loophole problem in some recent experimental claims to produce three-particle entanglement. The problem consists in the question whether mixtures of two-particle entangled states might suffice to explain the experimental data. In an attempt to close this loophole, we review two sufficient conditions that distinguish between N-particle states in which all N particles are entangled to each other and states in which only M particles are entangled ͑with M ϽN). It is shown that three recent experiments to obtain three-particle entangled states ͓Bouwmeester et al., Phys. Rev. Lett. 82, 1345 ͑1999͒; Pan et al., Nature 403, 515 ͑2000͒; and Rauschenbeutel et al., Science 288, 2024, ͑2000͔͒ do not meet these conditions. We conclude that the question whether these experiments provide confirmation of three-particle entanglement remains unresolved. We also propose modifications of the experiments that would make such confirmation feasible.