Abstract
One diagnosis of Bell’s theorem is that its premise of Outcome Independence is unreasonably strong, as it postulates one common screener system that purports to explain all the correlations involved. This poses a challenge of constructing a model for quantum correlations that is local, non-conspiratorial, and has many separate screener systems rather than one common screener system. In particular, the assumptions of such models should not entail Bell’s inequalities. Although we stop short of proving that such models exist (or do not exist), we obtain a few results relating them to models with a common screener system. We prove that if such a model exists, then there exists a local common screener system model for quantum correlations breaking Bell’s inequalities; that model, however, can be conspirational. We also show a way of transforming a model with separate screener systems for the Bell-Aspect correlations that satisfies strong Parameter Independence (PI) and No-Conspiracy (NOCONS) into a somewhat different model for the same correlations in which strong PI is somewhat compromised, but NOCONS and “regular” PI hold, and the elements of the constructed partition act deterministically with respect to measurement results. This means that such models allow the derivation of the Clauser-Horne-Shimony-Holt inequalities.
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Notes
- 1.
The stochastic version assumed probabilistic working of hidden states; there were other premises of the proof.
- 2.
- 3.
We frequently omit the “∩” sign between names of events.
- 4.
Especially since the following simple fact holds: let \(\langle \Omega,\mathcal{F},P\rangle\) be a probability space. Let \(\mathcal{G} =\big\{\{ A_{i},B_{i}\}\big\}_{i\in I} \subseteq \mathcal{F}^{2}\) be a finite family of pairs of correlated events in \(\mathcal{F}\). Then there exists a partition \(\mathcal{C}\) of \(\Omega \) such that for any \(C \in \mathcal{C}\) and for any i ∈ I, P(A i B i | C) = P(A i | C)P(B i | C). To construct this partition simply take all Boolean combinations of all correlated events, throwing out the empty ones should they arise.
- 5.
In some papers following Szabó’s work these conditions are called Locality, No Conspiracy, and Screening-off, resp.
- 6.
Szabó paper reports on his computer simulations aimed to construct such models.
- 7.
- 8.
By “Bell-Aspect experiment” we mean the experiment envisioned by Bell (1964) and famously carried out by Aspect et al. (1982), the essence of which we recall below. The correlations investigated by Aspect et al. we also call Bell-Aspect correlations. These correlations violate the so-called Clauser-Horne-Shimony-Holt inequalities.
- 9.
This is in essence definition 5 of a screener-off system of Hofer-Szabó (2008); in contrast to his definition, the screener system is defined here for a pair of events, regardless of whether or not they are correlated. In the sequel we frequently omit the brackets when speaking about correlated pairs of events.
- 10.
From now on, for two events X and Y, K(XY ) is a natural number being the size of the screener systems for these two events. We say “the” screener system even though of course many different screener systems may exist for some given events, but one particular will always be intended by the context. If we allow for an infinite screener system, we should understand by K(XY ) an index set of cardinality equal to the cardinality of the screener system and write α ∈ K(XY ) rather than k < K(XY ). If X = A i and Y = B j we will use the expression K(ij).
- 11.
We choose the single space rather then the many-space approach not because we prefer it (in fact we do not), but because it is employed in the majority of the literature on the subject of the connections between separate- and common common causes (or screener systems) and the Bell inequalities.
- 12.
Recall: an algebra of sets is atomic if for every non-minimal element p there exists an atom a such that a ⊆ p; an algebra of sets is atomistic if it is atomic and such that every non-minimal element is a union of atoms.
- 13.
If we allow for infinite screener systems, we should write α ∈ K(13), etc., as suggested in Footnote 10.
- 14.
This may happen e.g. when the correlation between A i m and B j n is perfect, i.e., when P(A i m B j n∣a i b j ) = 1. Even though such a case is experimentally unrealisable, we cater to it for more generality.
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Wroński, L., Placek, T., Godziszewski, M.T. (2017). Separate Common Causes and EPR Correlations: An “Almost No-Go” Result. In: Hofer-Szabó, G., Wroński, L. (eds) Making it Formally Explicit. European Studies in Philosophy of Science, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-55486-0_5
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