## Finite Statistics Loophole in CH, Eberhard, CHSH Inequalities

**Authors:** Justin Lee

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 "finite statistics" loophole is known to allow local realism to violate these inequalities if a sample size is small [1]. Remarkably though, this paper shows that this loophole in conjunction with an improper statistical analysis and incorrect singles counts can cause a violation of Eberhard inequality even with a large sample size, e.g. a 13 sigma violation was achieved despite 12,000,000 total trials in a Monte Carlo simulation of a local realist photonic experiment, and furthermore, a 27 sigma violation was produced when a small, acceptable 0.2% production rate loophole was applied. In order to properly analyze the data, a sample mean of Eberhard inequality values should be used to calculate the statistical strength, instead of using an aggregate Eberhard inequality value, and the correct singles counts should be used. Secondly, this paper shows that if a sample size does not far exceed the "large enough" value for the normal approximation of a Binomial distribution, it can still violate these inequalities, e.g. CHSH violation of 2:43 +/- 0:31 was achieved with 280 total trials and 2:16 +/- 0:13 even 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].

**Comments:** 16 Pages.

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### Submission history

[v1] 2018-03-06 01:44:19

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