Bell’s Theorem, Bell Inequalities, and the “Probability Normalization Loophole”

Guangye Chen

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.

Gordon Watson

Pierre A. Millette

John R. Dixon

We consider the Bell’s Theorem setup of Gill et. al. (2002). We present a “proof of concept” that if the source emitting the particles can predict the settings of the detectors with sufficiently large probability, then there is a scenario consistent with local realism that violates the Bell inequality for the setup.

Waldimar Amaya, Lynden K. Shalm, Sae Woo Nam +18 more

John Bell’s theorem of 1964 states that local elements of physical reality, existing independent of measurement, are inconsistent with the predictions of quantum mechanics (Bell, J. S. (1964), Physics (College. Park. Md). Specifically, correlations between measurement results from distant entangled systems would be smaller than predicted by quantum physics. This is expressed in Bell’s inequalities. Employing modifications of Bell’s inequalities, many experiments have been performed that convincingly support the quantum predictions. Yet, all experiments rely on assumptions, which provide loopholes for a local realist explanation of the measurement. Here we report an experiment with polarization-entangled photons that simultaneously closes the most significant of these loopholes. We use a highly efficient source of entangled photons, distributed these over a distance of 58.5 meters, and implemented rapid random setting generation and high-efficiency detection to observe a violation of a Bell inequality with high statistical significance. The merely statistical probability of our results to occur under local realism is less than 3.74×10-31, corresponding to an 11.5 standard deviation effect.

Lin Yicheng Kelvin