Incompleteness in the Bell Theorem with an Arbitrary Number of Settings

Abstract

We consider the Bell experiment for a system described by multipartite states in the case wherendichotomic observables are measured per site. Ifnis two, we consider a two-setting Bell experiment. IfnisM, we consider aM-setting Bell experiment. Two-setting model is an explicit local realistic model for the values of a correlation function, given in a two-setting Bell experiment.M-setting model is an explicit local realistic model for the values of a correlation function, given in aM-setting Bell experiment. Surprisingly we can discuss incompleteness in the Bell theorem. Also, we show a loophole problem of all the Bell-CHSH experimental claims. We discuss every Bell-CHSH experiment admits a local realistic theory if we rule out probability and statistics from the analysis of the experiment. We cannot obtain a violation of a Bell-CHSH inequality when we use only number theory and we do not use probability and statistics.