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AIP conference proceedings

Single, Complete, Probability Spaces Consistent With EPR-Bohm-Bell Experimental Data

David Avis, Paul Fischer, Astrid Hilbert, Andrei Yu. Khrennikov, Luigi Accardi, Guillaume Adenier, Christopher A. Fuchs, Gregg S. Jaeger, Andrei Yu. Khrennikov, Jan-Åke Larsson, Stig Stenholm

Published Jan 1, 2009• 21 citations
Publisher
DOI
10.1063/1.3109952

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We analyze anomalies in data to test the violation of Bell's inequality for the EPR-Bohm experiment. We found that the experimental correlations for photon polarization have an intriguing property. In the experimental data there are visible non-negligible deviations of probabilities $P_{++}^{\rm{exp}}(α, β), P_{+-}^{\rm{exp}}(α, β), P_{-+}^{\rm{exp}}(α, β), P_{--}^{\rm{exp}}(α, β) $ from the predictions of quantum mechanics, namely, $P_{++}(α, β)=P_{--}(α, β)= {1/2}\cos^2(α-β)$ and $P_{+-}=P_{-+}(α, β)={1/2}\sin^2(α-β).$ However, in some mysterious way those deviations compensate each other and finally the correlation $E^{\rm{exp}}(α, β)= P_{++}^{\rm{exp}}(α, β)- P_{+-}^{\rm{exp}}(α, β)- P_{-+}^{\rm{exp}}(α, β)+ P_{--}^{\rm{exp}}(α, β)$ is in the complete agreement with the QM-prediction, namely, $E(α, β)= P_{++}(α, β)- P_{+-}(α, β)- P_{-+}(α, β)+ P_{--}(α, β)= \cos 2(α-β).$ Therefore such anomalies play no role in the Bell's inequality framework. Nevertheless, other linear combinations of experimental probabilities do not have such a compensation property. There can be found non-negligible deviations from predictions of quantum mechanics. Thus neither classical nor quantum model can pass the whole family of statistical tests given by all possible linear combinations of the EPR-Bohm probabilities. Does it mean that both models are wrong?

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We show that paradoxical consequences of violations of Bell's inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely consistent probabilistic model by taking into account the probabilities of selecting the settings of the PBSs. Our model matches both the experimental data and is consistent with classical probability theory.
We analyze anomalies in data to test the violation of Bell's inequality for the EPR-Bohm experiment. We found that the experimental correlations for photon polarization have an intriguing property. In the experimental data there are visible non-negligible deviations of probabilities $P_{++}^{\rm{exp}}(α, β), P_{+-}^{\rm{exp}}(α, β), P_{-+}^{\rm{exp}}(α, β), P_{--}^{\rm{exp}}(α, β) $ from the predictions of quantum mechanics, namely, P++(α,β)=P−−(α,β)=1/2cos⁡2(α−β)P_{++}(α, β)=P_{--}(α, β)= {1/2}\cos^2(α-β)P++​(α,β)=P−−​(α,β)=1/2cos2(α−β) and P+−=P−+(α,β)=1/2sin⁡2(α−β).P_{+-}=P_{-+}(α, β)={1/2}\sin^2(α-β).P+−​=P−+​(α,β)=1/2sin2(α−β). However, in some mysterious way those deviations compensate each other and finally the correlation Eexp(α,β)=P++exp(α,β)−P+−exp(α,β)−P−+exp(α,β)+P−−exp(α,β)E^{\rm{exp}}(α, β)= P_{++}^{\rm{exp}}(α, β)- P_{+-}^{\rm{exp}}(α, β)- P_{-+}^{\rm{exp}}(α, β)+ P_{--}^{\rm{exp}}(α, β)Eexp(α,β)=P++exp​(α,β)−P+−exp​(α,β)−P−+exp​(α,β)+P−−exp​(α,β) is in the complete agreement with the QM-prediction, namely, E(α,β)=P++(α,β)−P+−(α,β)−P−+(α,β)+P−−(α,β)=cos⁡2(α−β).E(α, β)= P_{++}(α, β)- P_{+-}(α, β)- P_{-+}(α, β)+ P_{--}(α, β)= \cos 2(α-β).E(α,β)=P++​(α,β)−P+−​(α,β)−P−+​(α,β)+P−−​(α,β)=cos2(α−β). Therefore such anomalies play no role in the Bell's inequality framework. Nevertheless, other linear combinations of experimental probabilities do not have such a compensation property. There can be found non-negligible deviations from predictions of quantum mechanics. Thus neither classical nor quantum model can pass the whole family of statistical tests given by all possible linear combinations of the EPR-Bohm probabilities. Does it mean that both models are wrong?