Do All Pure Entangled States Violate Bell’s Inequalities for Correlation Functions?

Any pure entangled state of two particles violates a Bell inequality for two-particle correlation functions (Gisin’s theorem). We show that there exist pure entangled $N>\mathrm{}2\mathrm{}$ qubit states that do not violate any Bell inequality for $N$ particle correlation functions for experiments involving two dichotomic observables per local measuring station. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description.