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arXiv:1409.5158 · Journal of Advanced Physics

Analysis of the Christensen et al. Test of Local Realism

Donald A. Graft

Published Jun 3, 2015• 5 citations
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Abstract

The Clauser-Horne (CH) inequality can validly test aspects of locality when properly applied. This paper analyzes a recent CH-based EPRB experiment, the Christensen et al. experiment. Full details of the data analysis applied to the experiment are given. An alternative analysis is also presented that considers the role of accidental coincidences and confirms and justifies the main analysis. It is shown that the experiment confirms locality and disconfirms the quantum joint prediction. To make sense of this surprising finding, the conclusion presents a new rational interpretation of the EPR paradox. The paper also contributes to promulgation of robust and correct data analysis by describing the important degrees of freedom that affect the analysis, and that must be addressed in the analysis of any EPRB experiment.

DOI
10.1166/jap.2015.1198

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The Clauser-Horne (CH) inequality can validly test aspects of locality when properly applied. This paper analyzes a recent CH-based EPRB experiment, the Christensen et al. experiment. Full details of the data analysis applied to the experiment are given. An alternative analysis is also presented that considers the role of accidental coincidences and confirms and justifies the main analysis. It is shown that the experiment confirms locality and disconfirms the quantum joint prediction. To make sense of this surprising finding, the conclusion presents a new rational interpretation of the EPR paradox. The paper also contributes to promulgation of robust and correct data analysis by describing the important degrees of freedom that affect the analysis, and that must be addressed in the analysis of any EPRB experiment.
Thanks to its immunity to the detection loophole, the Clauser-Horne/Eberhard inequality plays an important role in tests of locality and in certification of quantum information protocols based on entanglement. I describe a local model that violates the inequality using a plausible mechanism relying upon a parameter of the apparatus, the source emission rate. The effect is generated through the analysis of time-tagged data using a standard windowed coincidence counting method. Significantly, the detection times here are not functions of the measurement settings, i.e., the fair coincidences assumption is satisfied. This finding has implications for the design and interpretation of experiments and for quantum information protocols, as it shows that the coincidence window mechanism cannot be eliminated by a demonstration of independence of the detection times and settings. The paper describes a reliable coincidence counting method and shows that it delivers an accurate count of true coincidences. Recent experimental tests of local realism based on the Clauser-Horne/Eberhard inequality are considered and it is shown that in one case (Christensen et al.) the emission rate is appropriately limited to ensure valid counting, and the data supports locality; in a second case (Giustina et al.) the experiment neglects to appropriately limit the emission rate, and the claimed violation can be accounted for locally.
A number of papers have suggested that it is inappropriate to combine data from different experiments when undertaking experimental tests of Bell's inequalities. It has been suggested that a correct analysis, using a single probability space, leads to inequalities which are not violated by experiment. If correct, this would be contrary to the normal interpretation of such experimental data. However, in this note, a generalised Clauser-Horne-Shimony-Holt (CHSH) inequality is derived for a system of four experiments constructed on a single probability space which combines the data from the four experiments. It is shown that this leads to the standard CHSH inequality which is normally used to interpret experimental data. Thus the commonly accepted conclusion that experimental violations of Bell's inequality imply that local realistic models are inconsistent with the predictions of quantum mechanics has not been challenged.
Recent experiments (Giustina et al 2013 Nature 497 227–30 ; Christensen et al 2013 Phys. Rev. Lett. 111 130406 ) have reached detection efficiencies sufficient to close the detection loophole, testing the Clauser–Horne version of Bell's inequality. For a similar future experiment to be completely loophole-free, it will be important to have discrete experimental trials with randomized measurement settings for each trial, and the statistical analysis should not overlook the possibility of a local state varying over time with possible dependence on earlier trials (the ‘memory loophole’). In this paper, a mathematical model for such an experiment is presented, and a method for statistical analysis that is robust to memory effects is introduced. Additionally, a new method for calculating exact p -values for martingale-based statistics is described; previously, only non-sharp upper bounds derived from the Azuma–Hoeffding inequality have been available for such statistics. This improvement decreases the required number of experimental trials to demonstrate non-locality. The statistical techniques are applied to the data of Giustina et al (2013 Nature 497 227–30 ) and Christensen et al (2013 Phys. Rev. Lett. 111 130406 ) and found to perform well.
Clauser-Horne (CH) inequality, Eberhard inequality, and Clauser-Horne-Shimony-Holt (CHSH) inequality are used to determine whether quantum entanglement can contradict local realism. However, the loophole is known to allow local realism to violate these inequalities if a sample size is small and not enough [1]. Remarkably though, this paper shows that this loophole can still cause a violation in these inequalities even with a very large sample size, e.g. a 2.4 sigma violation of CH inequality and Eberhard inequality was achieved despite 12,000,000 total trials in a Monte Carlo simulation of a local realist photonic experiment based on Malus' law. In addition, this paper shows how Eberhard inequality is especially vulnerable to this loophole when combined with an improper statistical analysis and incorrect singles counts, e.g. a 13.0 sigma violation was achieved with the same large sample size, and furthermore, a 26.6 sigma violation was produced when a small, acceptable 0.2% production rate loophole was applied. Supplementally, this paper demonstrates how the finite statistics loophole allows a bigger violation in a smaller sample size despite the sample size being enough, e.g. a CHSH violation of 4.4 sigma (2.43 +/- 0.10) was achieved with 280 total trials, and 4.0 sigma (2.16 +/- 0.04) with 3,000 total trials. This paper introduces the aforementioned loopholes as plausible local realist explanations to two observed violations reported by Giustina, et al. [2], and Hensen, et al. [3].
The assumption of local realism imposes constraints, such as Bell inequalities, on quantities obtained from measurements. In recent years, various tests of local realism have gained popularity in undergraduate laboratories, giving students the exciting opportunity to experimentally contradict this philosophical assumption. The standard test of the CHSH (Clauser–Horne–Shimony–Holt) Bell inequality requires 16 measurements, whereas a test of Freedman's inequality requires only three measurements. The calculations required to test Freedman's inequality are correspondingly simpler and the theory is less abstract. We suggest that students may benefit from testing Freedman's inequality before proceeding to the CHSH inequality and other more complicated experiments. Our measured data violated Freedman's inequality by more than six standard deviations.