Bell’s inequality for n spin-$s$ particles

The Mermin-Klyshko inequality for n spin-$\frac{1}{2}$ particles and two dichotomic observables is generalized to n spin-$s$ particles and two maximal observables. It is shown that some multiparty multilevel Greenberger-Horne-Zeilinger states [A. Cabello, Phys. Rev. A 63, 022104 (2001)] maximally violate this inequality for any $s.\mathrm{}$ For a fixed $n,$ the magnitude of the violation is constant for any $s,$ which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-$s$ particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed $s,$ the violation grows exponentially with $n,$ which provides a generalization to any s of Mermin’s conclusion for n $\mathrm{spin}-\hspace{0.5em}\frac{1}{2}$ particles [Phys. Rev. Lett. 65, 1838 (1990)].