Violation of local realism by a system withNspin-12particles

Recently, it was found that Mermin’s inequalities may not always be optimal for the refutation of a local realistic description [Phys. Rev. Lett. 88, 210402 (2002)]. To complete this work, we derive an inequality for the Greenberger-Horne-Zeilinger-type pure state for a system with N spin-$\frac{1}{2}$ particles and the violation of the inequality can be shown for all the non product pure states. Mermin’s inequality for a system of N spin-$\frac{1}{2}$ particles and Gisin’s theorem for a system of two spin-$\frac{1}{2}$ particles are both included in our inequality.