Sufficient conditions for three-particle entanglement and their tests in recent experiments

We review three conditions that distinguish between states of $N$ particles in which all $N$ particles are entangled to each other, and states in which only $M$ particles entangled (with $M<N$). These conditions are used to analyse recent experiments to obtain three-particle entangled states (Bouwmeester et al. Phys. Rev. Lett. {\bf 82}, 1345 (1999), Pan et al. Nature {\bf 403}, 515 (2000), and Rauschenbeutel et al. Science, {\bf 288}, 2024, (2000)). It is shown that these experiments do not meet the three conditions discussed here and can therefore not yet be considered as definitive confirmation of three-particle entanglement. We also discuss modifications of the experiments which would make such confirmation possible.