Hardy Relations and Common Cause
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Hardy Relations and Common Cause Katsuaki Higashi1 Received: 6 April 2020 / Accepted: 5 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Some researchers argued that in the non-existence proof of hidden variables, the existence of a common common-cause of multiple correlations is tacitly assumed and that the assumption is unreasonably strong. According to their idea, it is sufficient if the separate common-cause of each correlation exists. However, for such an idea, various no-go results are already known. Recently, Higashi showed that there exists no local separate common-cause model for the correlations that appear in Hardy’s famous argument. In this paper, I give another simple and suggestive proof of the same content. First, I will show that there exists no local common commoncause model of the correlations that appear in Hardy’s argument. Second, taking the proof as a hint, following almost the same steps, I will show the non-existence of a local separate common-cause model for those correlations. Finally, based on the argument in the previous sections, I will discuss what we can conclude about the issue of reducibility from a separate common-cause model to a common commoncause model. It will be concluded that it is “irreducible” at least by a usual method. Keywords Hardy’s argument · Common cause · Separate common cause · Reducibility
1 Introduction The conventional view is that there exists no common cause of quantum correlations (e.g. [1, 2]). Because, when we assume the existence of a common cause, Bell’s inequalities are derived and the violation of the inequalities has been repeatedly confirmed by the experiments (e.g. [3]). On the other hand, some researchers [4–6] argued that in the derivations of Bell’s inequalities, it is tacitly assumed that there exists a common common-cause in multiple correlations and the assumption is a too strong requirement. They consider what is needed to explain the quantum * Katsuaki Higashi higashi.katsuaki@nihon‑u.ac.jp 1
Faculty of Marine Technology, Tokyo University of Marine Science and Technology, 2‑1‑6, Etchujima, Koto‑ku, Tokyo 135‑8533, Japan
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Foundations of Physics
mechanical correlation is a separate common-cause for each correlation. Although this idea is very interesting, various no-go theorems for separate common-cause approaches are already known [7–14]. Graßhoff et al. [7] showed that there exists no local separate common-cause model in the maximally entangled states. In addition, Higashi [14] recently used Hardy’s famous argument to show that, in any non-maximally entangled state, there exists no local separate common-cause model. Therefore, it became clear that, in any entangled state, there exists no local separate common-cause model. In this paper, I present another simple and suggestive proof of the mathematical fact which Higashi proved (see Fact 6 in [14]). First, in Sect. 3.1, I will show that there exists no local common common-cause model for multiple correlations that app
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