Generalized spin-squeezing inequalities inN-qubit systems: Theory and experiment

We present detailed derivations, various improvements and application to concrete experimental data of spin squeezing inequalities formulated recently by some of us Phys. Rev. Lett. 95, 120502 (2005). These inequalities generalize the concept of the spin squeezing parameter and provide necessary and sufficient conditions for genuine two-, or three-qubit entanglement for symmetric states, and sufficient condition for general N-qubit states. We apply our method to theoretical study of Dicke states, and, in particular, to W states of N qubits. Then, we analyze the recently experimentally generated seven- and eight-ion W states Nature 438, 643 (2005). we also present some details concerning this experiment. Finally, we improve criteria for detection of genuine tripartite entanglement based on entanglement witnesses.