Generalized spin-squeezing inequalities in N-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.