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arXiv:0710.1935 · Journal of Physics A Mathematical and Theoretical

Classification of local realistic theories

Koji Nagata

Published Apr 2, 2008• 7 citations
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Abstract

Recently, it has been shown that an explicit local realistic model for the values of a correlation function, given in a two-setting Bell experiment (two-setting model), works only for the specific set of settings in the given experiment, but cannot construct a local realistic model for the values of a correlation function, given in a continuous-infinite settings Bell experiment (infinite-setting model), even though there exist two-setting models for all directions in space. Hence, the two-setting model does not have the property which the infinite-setting model has. Here, we show that an explicit two-setting model cannot construct a local realistic model for the values of a correlation function, given in an only discrete-three settings Bell experiment (three-setting model), even though there exist two-setting models for the three measurement directions chosen in the given three-setting experiment. Hence, the two-setting model does not have the property which the three-setting model has.

DOI
10.1088/1751-8113/41/15/155308

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Recently, it has been shown that an explicit local realistic model for the values of a correlation function, given in a two-setting Bell experiment (two-setting model), works only for the specific set of settings in the given experiment, but cannot construct a local realistic model for the values of a correlation function, given in a continuous-infinite settings Bell experiment (infinite-setting model), even though there exist two-setting models for all directions in space. Hence, the two-setting model does not have the property which the infinite-setting model has. Here, we show that an explicit two-setting model cannot construct a local realistic model for the values of a correlation function, given in an only discrete-three settings Bell experiment (three-setting model), even though there exist two-setting models for the three measurement directions chosen in the given three-setting experiment. Hence, the two-setting model does not have the property which the three-setting model has.
We review the historical fact of multipartite Bell inequalities with an arbitrary number of settings. An explicit local realistic model for the values of a correlation function, given in a two-setting Bell experiment (two-setting model), works only for the specific set of settings in the given experiment, but cannot construct a local realistic model for the values of a correlation function, given in a continuous-infinite settings Bell experiment (infinite-setting model), even though there exist two-setting models for all directions in space. Hence, the two-setting model does not have the property that the infinite-setting model has. Here, we show that an explicit two-setting model cannot construct a local realistic model for the values of a correlation function, given in an M -setting Bell experiment ( M -setting model), even though there exist two-setting models for the M measurement directions chosen in the given M -setting experiment. Hence, the two-setting model does not have the property that the M -setting model has.
Here, we consider the Bell experiment for a system described by multipartite states in the case where ndichotomic observables are measured per site.If n is two, we consider a two-setting Bell experiment.If n is infinite, we consider an infinite-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.Infinite-setting model is an explicit local realistic model for the values of a correlation function, given in an infinite-setting Bell experiment.In the non-contextual scenario, there is not the essential difference between infinite-setting model and two-setting model.And we cannot classify local realistic theories in this case.This says that we can construct infinite-setting model from two-setting model.Surprisingly we can discuss incompleteness in the Bell theorem by using non-contextual models.On the other hand, in the contextual scenario, there is the essential difference between infinite-setting model and two-setting model.This says that we must distinguish infinite-setting model from two-setting model.And we can classify local realistic theories in this case.
We have considered a two-particle Bell experiment to visualize the conflict between rotational invariance of physical laws and a specific local realistic theory. The experiment is reproducible by using a local realistic theory obtained in a two-setting Bell experiment. The generalized Bell inequality {[J. Phys. A: Math. Theor. {\bf 40}, 13101 (2007)]}, which is derived under the assumption that there exists a rotationally invariant local realistic theory, turns out to disprove such a local realistic model existing with a two-setting experiment. This implies that such a model is not rotationally invariant and should, therefore, be ruled out in some situations.
Bell’s theorem founded on Bell’s inequalities asserts that no local realistic theories can reproduce all quantum mechanical predictions for spin correlation of particle pairs. It is pointed out that the assumption of setting-independent probability makes Bell’s inequalities not impose constraint on all local realistic models and thus constitutes a theoretical loophole of Bell’s theorem. A counterexample is presented by showing that a local realistic model with appropriate probability density reproduces all quantum mechanical predictions. It becomes clear that experiments violate Bell’s inequalities because the real correlation functions are not constrained by these inequalities. It is also exposed that, rigorous logical reasoning of counter factual deduction does not permit to exclude any premises of Bell’s inequalities by a certain amount of experimental violations of these inequalities.