Criteria to detect genuine multipartite entanglement using spin measurements

We derive conditions in the form of inequalities to detect the genuine $N$-partite entanglement of $N$ systems. The inequalities are expressed in terms of variances of spin operators, and can be tested by local spin measurements performed on the individual systems. Violation of the inequalities is sufficient (but not necessary) to certify the multipartite entanglement, and occurs when a type of spin squeezing is created. The inequalities are similar to those derived for continuous-variable systems, but instead are based on the Heisenberg spin-uncertainty relation $\Delta J_{x}\Delta J_{y}\geq|\langle J_{z}\rangle|/2$. We also extend previous work to derive spin-variance inequalities that certify the full tripartite inseparability or genuine multi-partite entanglement among systems with fixed spin $J$, as in Greenberger-Horne-Zeilinger (GHZ) states and W states where $J=1/2$. These inequalities are derived from the planar spin-uncertainty relation $(\Delta J_{x})^{2}+(\Delta J_{y})^{2}\geq C_{J}$ where $C_{J}$ is a constant for each $J$. Finally, it is shown how the inequalities detect multipartite entanglement based on Stokes operators. We illustrate with experiments that create entanglement shared among separated atomic ensembles, polarization-entangled optical modes, and the clouds of atoms of an expanding spin-squeezed Bose-Einstein condensate. For each example, we give a criterion to certify the mutual entanglement.