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 common-cause 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 common-cause model. It will be concluded that it is “irreducible” at least by a usual method.
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Notes
It is not necessary to limit to the non-maximally entangled state.
This idea of requiring a common cause originated from Reichenbach [19].
\(\{C_{k}\}_{k \in K}\) is common screening-off factors for the two correlations, so the superscript of screening-off factors in Definition 1 is omitted.
In Higashi [14], this requirement is stated in a different form (called \(\mathcal {C}\)-independence) however, they are mathematically equivalent.
To be exact, the screening-off factors are not always a logical partition. However, even in such a case, if one event having probability 0 is added, the logical partition is obtained.
The proof of Fact 3 does not use the distinction between maximality and non-maximality. This proof gives an explanation as to why Hardy’s paradox does not occur in a maximally entangled state. As can be easily seen, in a maximally entangled state, for example, when B2 holds, \(Corr (L_ {1+}, R_ {2+})\) is a perfect correlation. However, then, from fact 3, the Hardy relations cannot be established.
However, as they themselves know, they assume a stronger condition than S2.
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Higashi, K. Hardy Relations and Common Cause. Found Phys 50, 1382–1397 (2020). https://doi.org/10.1007/s10701-020-00392-y
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DOI: https://doi.org/10.1007/s10701-020-00392-y