Multipartite Bell-type inequality by generalizing Wigner's argument

Wigner's argument inferring a Bell-type inequality for the Einstein–Podolsky–Rosen–Bohm entangled state is generalized here for any $N$-partite state. This is based on assuming for the relevant dichotomic observables the existence of the overall joint probability distributions, satisfying the locality condition, which would yield the measurable marginal probabilities. For any $N$, such a generalized Wigner inequality (GWI) is violated by quantum mechanics for all pure entangled states. The efficacy of GWI is probed, comparing with the Seevinck–Svetlichny multipartite Bell-type inequality, by calculating threshold visibilities for the quadripartite Greenberger–Horne–Zeilinger, Cluster, and $W$ states that determine their respective robustness with respect to the quantum-mechanical violation of GWI in the presence of white noise.