Bell's theorem for trichotomic observables

Bell's paradoxes, due to the fundamental properties of light and the nature of the photon, are discussed within a single framework with a view to checking the hypothesis that a stationary, non-negative, joint probability distribution function exists. This hypothesis, related to the local theory of hidden parameters as a possible interpretation of quantum theory, enables experimentally verifiable Bell's inequalities to be formulated. The dependence of these inequalities on the number of observers $V$ is considered. Quantum theory predicts the breakdown of Bell's inequalities in optical experiments. It is shown that as $V$ increases, the requirements on the quantum effectiveness of the detector, $\eta$, are reduced from $\eta>2/3$ at $V$=2 to $\eta>1/2$ for $V\to\infty$. Examples of joint probability distribution functions are given for illustrative purposes, and a way to resolve the Greenberg–Horne–Zeilinger (GHZ) paradox is suggested.