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A Note on Bell’s Theorem Logical Consistency

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Abstract

Counterfactual definiteness is supposed to underlie the Bell theorem. An old controversy exists among those who reject the theorem implications by rejecting counterfactual definiteness and those who claim that, since it is a direct consequence of locality, it cannot be independently rejected. We propose a different approach for solving this contentious issue by realizing that counterfactual definiteness is an unnecessary and inconsistent assumption. Counterfactual definiteness is not equivalent to realism or determinism neither it follows from locality. It merely reduces to an incongruent application of counterfactual reasoning. Being incompatible with falsifiability, it constitutes an unjustified assumption that goes against the scientific method rigor. Correct formulations of the Bell theorem’s bases show it is absent either as a fundamental hypothesis or as a consequence of something else. Most importantly, we present a coherent Bell inequality derivation carefully devised to show explicitly and convincingly the absence of incompatible experiments or counterfactual reasoning. Thus, even admitting that counterfactual definiteness could be a consistent assumption, the necessary conclusion is that it is irrelevant for the inequality formulation and can be safely ignored when discussing Bell’s inequality philosophical and physical implications.
Vol.:(0123456789)
Foundations of Physics (2021) 51:84
https://doi.org/10.1007/s10701-021-00488-z
1 3
A Note onBell’s Theorem Logical Consistency
JustoPastorLambare1 · RodneyFranco1
Received: 9 February 2021 / Accepted: 20 July 2021 / Published online: 31 July 2021
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature
2021
Abstract
Counterfactual definiteness is supposed to underlie the Bell theorem. An old con-
troversy exists among those who reject the theorem implications by rejecting coun-
terfactual definiteness and those who claim that, since it is a direct consequence of
locality, it cannot be independently rejected. We propose a different approach for
solving this contentious issue by realizing that counterfactual definiteness is an
unnecessary and inconsistent assumption. Counterfactual definiteness is not equiv-
alent to realism or determinism neither it follows from locality. It merely reduces
to an incongruent application of counterfactual reasoning. Being incompatible with
falsifiability, it constitutes an unjustified assumption that goes against the scientific
method rigor. Correct formulations of the Bell theorem’s bases show it is absent
either as a fundamental hypothesis or as a consequence of something else. Most
importantly, we present a coherent Bell inequality derivation carefully devised to
show explicitly and convincingly the absence of incompatible experiments or coun-
terfactual reasoning. Thus, even admitting that counterfactual definiteness could
be a consistent assumption, the necessary conclusion is that it is irrelevant for the
inequality formulation and can be safely ignored when discussing Bell’s inequality
philosophical and physical implications.
Keywords Bell inequality· Counterfactual definiteness· Falsifiability· Realism·
Locality
* Justo Pastor Lambare
jupalam@gmail.com
Rodney Franco
francorodney03@gmail.com
1 Facultad de Ciencias Exactas y Naturales, Ruta Mcal. J. F. Estigarribia, Km 11 Campus de la
UNA, SanLorenzo, Paraguay
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... Probably the most popular version is the CHSH inequality [17]. References [18][19][20][21][22] discuss the CHSH inequality correct and incorrect derivations warning against the use of counterfactual reasoning. The derivation of the Bell inequality requires only elementary mathematics and is uncontroversial from the mathematical point of view. ...
... They are unnecessary because, as we have seen in sect. 3.2, they are employed neither to prove quantum nonlocality nor to derive the Bell inequality [22]. ...
... Therefore, a coherent rejection of realism as a "causal explanation" has nothing to do with the infamous elements of physical reality or logically ill-conceived machinations such as counterfactual definiteness, incompatible experiments, or joint probabilities [22]. ...
Preprint
http://dx.doi.org/10.20944/preprints202205.0015.v1
... In their paper in Foundation of Physics [6], Lambare and Franco make some misleading statements, which we want to rectify. They correctly claim, that there is no need for a counterfactual reasoning in the context of Bell inequalities and explain , that LRHVM , which they call BDM, may be derived using local causality (LC), perfect correlations and measurement independence (MI). ...
... It is clear, that the frequentist poof of CHSH , given in [6], fails, if MI fails. Lambare et Franco realize this, but they believe, as many do, that the violation of MI, would mean "conspiracy" or "super-determinism", thus they dismiss such solution. ...
... We have 4 incompatible experiments, labelled by (x, y), and only 2 outcomes are outputted in each trial, thus JP of 4 random variables (A 1 , A -1 , B 1 , B -1 ) does not exist. It is easy to evaluate 4 expectations [6] entering the inequality (7) . ...
Preprint
In their recent paper, Lambare and Franco correctly claim that Bell deterministic model and inequalities may be derived using only local causality, perfect correlations and measurement independence, without talking about joint probabilities. However, they do not understand and they are not alone, that measurement independence has nothing to do with freedom of choice and no conspiracy. Measurement independence should be called noncontextuality, because it allows implementing random variables, describing incompatible random experiments, on a unique probability space, on which they are jointly distributed. Such implementation defines a probabilistic coupling, which we explain in detail in this paper. Their frequentist proof fails, if the probabilistic coupling and joint probabilities do not exist. We construct a probabilistic coupling for their counterexample to prove, that there is no contradiction with Fine Theorem. Nobody questions Bell Theorem logical consistency and nobody claims that Fine disproved Bell Theorem. Various metaphysical assumptions, such as local realism, classicality or counterfactual definiteness may motivate a choice of a probabilistic model. However, once a model is chosen, its meaning and its implications may only be discussed rigorously in a probabilistic framework. Bell inequalities are violated in various Bell Tests, for us, it proves that that hidden variables depend on settings confirming contextual character of quantum observables and an active role played by measuring instruments. Bell was a realist, thus he thought that he had to choose between nonlocality and super-determinism. From two bad choices he chose nonlocality. Today he would probably choose contextuality.
... Reference [41] presents two theorems proving the inconsistency of the counterfactual definiteness assumption. Reference [42] contains a derivation making explicit that exclusively predictions of actual experiments -therefore realizable -are involved in the derivation of the Bell inequality. ...
... It is relevant to note that the hidden variables approach does not use counterfactual reasoning [42]. As originally formulated, it is a no-go theorem for local hidden variables. ...
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We present a pragmatic analysis of the different meanings assigned to the term “local realism” in the context of the empirical violations of Bell-type inequalities since its inception in the late 1970s. We point out that most of them are inappropriate and arise from a deeply ingrained prejudice that originated in the celebrated 1935 paper by Einstein-Podolski-Rosen. We highlight the correct connotation that arises once we discard unnecessary metaphysics.
... RDG initiated this project and wrote an initial draft which he shared with JLP. It turned out that JLP had already written a critique on another work of Kupczynski together with Rodney Franco [9]. Discussions led to many changes and to the appendix connecting the results to further material in [1]. ...
... Section "Joint Probabilites" in Ref. [9] discusses other logical issues regarding Kupczynski's claims on the existence of joint probabilities. ...
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In a sequence of papers, Marian Kupczynski has argued that Bell's theorem can be circumvented if one takes correct account of contextual setting-dependent parameters describing measuring instruments. We show that this is not true. Taking account of such parameters in the way he suggests, the Bell-CHSH inequality can still be derived. Violation thereof by quantum mechanics cannot be easily explained away: quantum mechanics and local realism (including Kupczynksi's expanded concept of local realism) are not compatible with one another. Further inspection shows that Kupczynski is actually falling back on the detection loophole. Since 2015, numerous loophole-free experiments have been performed, in which the Bell-CHSH inequality is violated, so despite any other possible imperfections of such experiments, Kupczynski's escape route for local realism is not available.
... Reference [41] presents two theorems proving the inconsistency of the counterfactual definiteness assumption. Reference [42] contains a derivation making explicit that exclusively predictions of actual experiments -therefore realizable -are involved in the derivation of the Bell inequality. ...
... It is relevant to note that the hidden variables approach does not use counterfactual reasoning [42]. As originally formulated, it is a no-go theorem for local hidden variables. ...
Preprint
We present a pragmatic analysis of the different meanings assigned to the term "local realism'' in the context of the empirical violations of Bell-type inequalities since its inception in the late 1970s. We point out that most of them are inconsistent and arise from a deeply ingrained prejudice that originated in the celebrated 1935 paper by Einstein-Podolski-Rosen. We highlight the correct connotation that arises once we discard unnecessary metaphysics.
... Trying to give a direct physical interpretation to the sum appearing under the same integral sign leads to inconsistencies and is a frequent source of confusion, as discussed in Refs. [4][5][6][7]. ...
... The expression A k = α i β j should not be considered a functional factorization since α i and β j represent constants used to classify the different eigenvalues according to the enumeration scheme(7). ...
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In a recent article, Cetto et al. (Found Phys 50:27–39, 2020) present an elegant formalism analyzing the probabilistic properties of the quantum singlet state correlations. From their study, they conclude the quantum formalism entails a partitioning of the probability space which is supposed to be absent in Bell’s hidden variables probabilistic model. We formally prove that this is not the case and that Bell’s probabilistic model does indeed possesses the characteristic that is supposed to be missing. Therefore, contrary to their claim, their observation does not put into question the applicability of Bell-type inequalities to the bipartite singlet state.
... Likewise, when analyzing (4), operators' commutativity is not an issue. According to (3) No matter what other inequality interpretations we may find, they do not invalidate the locality implications of a CHSH singlet state correlation experiment [5,10]. ...
... At least when it is correctly formulated. Notice the absence of counterfactual definiteness, see Refs.[5,6] ...
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The view exists that the Bell inequality is a mere inconsistent application of classical concepts to a well established quantum world. In the article, ``Nonlocality claims are inconsistent with Hilbert-space quantum mechanics'' [Phys. Rev. A, 101, 022117, (2020)] Robert B. Griffiths advocates for the locality of quantum theory. Although R. B. Griffiths presents valuable insights in favor of quantum mechanics' local character, he based some of them on unjustified views concerning the Bell inequality interpretation.
... Refs. [4,5] explain similar inconsistencies arising from joint probabilities and incompatibility. ...
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Marian Kupczynski(MK)is the author of a controversial paper published (2020) in the journal Frontiers in Physics. The work is built around a mathematical claim by MK which is actually false, and MK's logical reasoning around his claim is also incorrect. The same claim was made by him in several other recent papers published in other journals. A proof that the claimed result is false is the main content of our present "Comment". It is purely a mathematical counter-example to a mathematical claim in a number of MK's papers.
... Members of so called probabilistic opposition, who are indirectly criticised by G-L in [1] and by Lambare and Franco in [9], arrived ,often independently, to the same correct conclusions. ...
Preprint
In a recent preprint Gill and Lambare, criticize our paper published in Frontiers in Physics. Their criticism is unfounded and misleading. They define a probabilistic coupling, in which BI-CHSH hold for all finite samples. It does not mean, that BI-CHSH hold in our model, in which four incompatible experiments are described by setting dependent random variables implemented on 4 disjoint dedicated probability spaces. A joint probability distribution of these random variables does not exist and may not be used to derive inequalities. Moreover, their probabilistic coupling is useless, for a subsequent contextual model, which we construct to describe final data from Bell tests and to explain, in a locally causal way, the reported violations of inequalities and apparent violations of no-signaling. Neither quantum probabilistic model of an ideal EPRB experiment nor local realistic and stochastic hidden variable models may explain reported non-signaling Therefore; it is obvious that our model extends the set of probability distributions of possible measurements allowed in the standard hidden variable models. Gill and Lambare seem not understand , the main message of our paper, that the violation of BI-CHSH and Eberhard inequalities by finite samples in Bell Tests, no matter how well these tests are designed and performed, does not allow for doubt regarding the existence of objective external physical reality and causal locality in Nature. Our contextual model does not want to circumvent Bell Theorem. Therefore the title of Gill and Lambare paper and the conclusion: Kupczynski's escape route for local realism is not available are misleading and have nothing to do with the content and conclusions of our paper.
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