Bell's theorem states that no description of a Bell experiment can be simultaneously local, realistic in the sense of counterfactual definiteness, and free of conspiracy between settings and hidden state. The recent generation of experiments has confirmed the predicted violation of the CHSH inequality, so one of the assumptions must be abandoned. Which one, and how one reconstructs a coherent worldview after doing so, is a question on which many authors disagree. This paper is written by three such authors. All three reject both counterfactual definiteness and conspiratorial violation of statistical independence of setting choices and state. After a joint exposition of the classical half of Bell's theorem in the language of Pearl-style causal graphs, a joint summary of the loophole-free experiments, and a joint survey of the recent literature, each author states where they have presently arrived. Gill accepts irreducible and non-local quantum randomness and finds the choice between locality and realism a false dichotomy. In his later works, Bell derives counterfactual definiteness from classical local causality, and that is what has to go. The metaphysical concepts "realism", "locality", "causality" need to be reconsidered. Helland reconstructs the Hilbert-space formalism from a theory of accessible variables, and from this theory he concludes that every observer must be limited in a specific sense. Jongejan proposes a geometric hidden-variable construction in which the degree of violation of the CHSH inequality depends on the number of dimensions of space, Tsirelson's bound corresponding to three dimensions. The authors conclude with a discussion.
Bell's theorem states that no description of a Bell experiment can be simultaneously local, realistic in the sense of counterfactual definiteness, and free of conspiracy between settings and hidden state. The recent generation of experiments has confirmed the predicted violation of the CHSH inequality, so one of the assumptions must be abandoned. Which one, and how one reconstructs a coherent worldview after doing so, is a question on which many authors disagree. This paper is written by three such authors. All three reject both counterfactual definiteness and conspiratorial violation of statistical independence of setting choices and state. After a joint exposition of the classical half of Bell's theorem in the language of Pearl-style causal graphs, a joint summary of the loophole-free experiments, and a joint survey of the recent literature, each author states where they have presently arrived. Gill accepts irreducible and non-local quantum randomness and finds the choice between locality and realism a false dichotomy. In his later works, Bell derives counterfactual definiteness from classical local causality, and that is what has to go. The metaphysical concepts "realism", "locality", "causality" need to be reconsidered. Helland reconstructs the Hilbert-space formalism from a theory of accessible variables, and from this theory he concludes that every observer must be limited in a specific sense. Jongejan proposes a geometric hidden-variable construction in which the degree of violation of the CHSH inequality depends on the number of dimensions of space, Tsirelson's bound corresponding to three dimensions. The authors conclude with a discussion.
In this short note, I derive the Bell-CHSH inequalities as an elementary result in the present-day theory of statistical causality based on graphical models or Bayes' nets, defined in terms of DAGs (Directed Acyclic Graphs) representing direct statistical causal influences between a number of observed and unobserved random variables. I show how spatio-temporal constraints in loophole-free Bell experiments, and natural classical statistical causality considerations, lead to Bell's notion of local hidden variables, and thence to the CHSH inequalities. The word “local” applies to the way that the chosen settings influence the observed outcomes. The case of contextual setting-dependent hidden variables (thought of as being located in the measurement devices and dependent on the measurement settings) is automatically covered, despite recent claims that Bell's conclusions can be circumvented in this way.
Bell's theorem is popularly stated as the inconsistency of quantum theory with local realism. Realism is here formalized by the ontological models framework, where elements of reality are points of a measurable space and probabilities obey the Kolmogorov axioms. In the Bell scenario, locally causal ontological models posit a common cause supporting a global valuation, a joint assignment of outcomes to all measurements at once, across incompatible contexts. Within this notion of realism, Bell's theorem makes any realist explanation of the quantum predictions conspiratorial: any mechanism, whether superluminal, superdeterministic, or retrocausal, must be finely tuned so as never to manifest operationally. We redefine realism through a new ontological framework that generalizes the standard one. The elements of reality, which we call records, are no longer points but relational parts of a point-free generalization of a topological space (sublocales of a locale). A record shared by two incompatible contexts establishes the sameness of a measurement across them, without such pairwise identities composing into a global valuation, and the common cause is then a web of such records. Within this framework, Bell's no-go theorem becomes a go theorem: local causality, restated for records, is no longer at odds with the quantum statistics. This is possible because modularity, the inclusion-exclusion law of classical probability, fails across incompatible contexts, so that the probability theory generalizes Kolmogorov's. We exhibit explicit models reproducing several quantum strategies, including the optimal CHSH violation. More broadly, locally causal models of this kind attain every Bell-CHSH value up to the algebraic maximum. A realist explanation of the quantum predictions need not be conspiratorial; every causal influence posited by these models is reflected in the operational predictions.
Chapter 7 discusses the Bell theorem and its demonstration using Mermin’s machine. The Bell theorem proves that a theory based on local realism (one with hidden variables) must satisfy certain inequalities; quantum mechanics predicts these are violated. Mermin designed a gedanken experiment having a double-particle source and two detectors each having three settings, which can correspond to measurements along three spin orientations. A simple hidden-variable assumption is shown to violate quantum predictions. The CHSH inequality, a form of the general Bell inequality, is proved. This relation has been shown to be violated experimentally in quantum experiments, implying that quantum mechanics indeed has “spooky action at a distance.” Many experiments over the years had loopholes, ways used to explain away the data in order to preserve local reality, but recent experiments have been able to eliminate most of these.
Loophole-free violations of Bell inequalities imply that at least one of the assumptions behind local hidden-variable theories must fail. Here, we show that, if only one fails, then it has to fail completely, therefore excluding models that partially constrain freedom of choice or allow for partial retrocausal influences, or allow partial instantaneous actions at a distance. Specifically, we show that (i) any hidden-variable theory with outcome independence (OI) and arbitrary joint relaxation of measurement independence (MI) and parameter independence (PI) can be experimentally excluded in a Bell-like experiment with many settings on high-dimensional entangled states, and (ii) any hidden-variable theory with MI, PI and arbitrary relaxation of OI can be excluded in a Bell-like experiment with many settings on qubit-qubit entangled states. Loophole-free violations of Bell inequalities imply that at least one of the assumptions formalising local hidden-variable theories must fail. Here, the authors show that, if only one is false, then it has to fail completely, therefore excluding models that allow partial instantaneous actions at a distance, partially constrain freedom of choice or allow for partial retrocausal influences.
We argue that for the proof of Bell's theorem no assumptions about realism or free will are necessary. The key formula E(AB∣a,b)=∫A(a,b,λ)B(a,b,λ)ρ(λ)dλ follows from the logic of plausible reasoning (the objective Bayesian interpretation of probability theory) taken alone, without any further assumptions about realism. The space Λ, usually interpreted as some space of `hidden variables', can be constructed for an arbitrary `field of discourse' using Stone's theorem. The rejection of superdeterminism follows from logical independence -- the non-existence of information which suggest a dependence -- of the free decisions of the experimenters from everything else. To prove the Bell inequality it is, then, sufficient to reduce this to E(AB∣a,b)=∫A(a,λ)B(b,λ)ρ(λ)dλ. This follows for space-like separated measurements from Einstein causality alone. So, the consequence of the violation of Bell's inequality is that Einstein causality has been empirically falsified, without any loopholes left. We consider and reject the idea of weakening the logic of plausible reasoning could be used to create a loophole. Finally, we discuss what can be used to replace Einstein causality. While weak (signal) Einstein causality holds, it follows either from a notion of strong causality or from human inability to prepare states which violate quantum uncertainty, which makes it uninteresting for fundamental considerations. A notion of strong causality has either unacceptable causal loops or requires a hidden preferred frame. Given that a hidden preferred frame is completely compatible with modern physics, it is argued that this choice is preferable.
Superdeterminism—where the Measurement Independence assumption in Bell’s Theorem is violated—is frequently assumed to imply implausibly conspiratorial correlations between properties λ of particles being measured and measurement settings x and y. But it does not have to be so: a superdeterministic but non-conspiratorial locally causal model is developed where each pair of entangled particles has unique λ. The model is based on a specific but arbitrarily fine discretisation of complex Hilbert space, where λ defines the information, over and above the freely chosen nominal settings x and y, which fixes the exact measurement settings X and Y of a run of a Bell experiment. Pearlean interventions, needed to assess whether x and y are Bell-type free variables, are shown to be inconsistent with rational-number constraints on the discretised Hilbert states. These constraints limit the post-hoc freedom to vary x keeping λ and y fixed but disappear with any coarse-graining of λ, X, and Y, rendering so-called drug-trial conspiracies irrelevant. Points in the discretised space can be realised as ensembles of symbolically labelled deterministic trajectories on an ‘all-at-once’ fractal attractor. It is shown how quantum mechanics might be ‘gloriously explained and derived’ as the singular continuum limit of the discretisation of Hilbert space. It is argued that the real message behind Bell’s Theorem has less to do with locality, realism, or freedom to choose, and more to do with the need to develop more explicitly holistic theories when attempting to synthesise quantum and gravitational physics.