Bell's inequality, trichotomic observables, and supplementary assumptions

A general approach to Bell-type inequalities for trichotomic observables is presented. With this formalism, which explicitly includes no-counts events, we show that the limits of all Bell-type inequalities are functions of the supplementary assumptions. This approach enables us to include in the causal and local theory the same assumptions used in the quantum-mechanical analysis of an experiment, and to deduce the inequality that must be satisfied by any local realistic model incorporating these assumptions. We prove that, using the quantum-mechanical assumption of ''random nondetection'' in the causal and local approach, a generalized Bell-type inequality is deduced whose upper limit can be violated by the quantum-mechanical predictions for symmetrical experimental setups involving detectors with quantum efficiency greater than 0.811.