Abstract Quantum theory has accounted for a wide range of physical effects in systems ranging from microscopic to macroscopic. However, it is associated with various paradoxes—such as the well-known EPR example. Einstein’s suggestion that the theory was incomplete has led to numerous attempts to underpin quantum theory by a hidden variable theory, including the Bell approach. This approach was originally based on certain assumptions: (i) realism (ii) locality and (iii) non-contextuality. In its developed form, the theory involved only probabilities for outcome values of system observables. For systems involving two sub-systems, the hidden variable theory predicted that correlations between measurement outcomes of observable pairs of from different sub-systems would satisfy certain—such as the CHSH Bell inequality. The inequality was inconsistent with quantum theory predictions and with experiment. Hence, the question arose: Which of the three assumptions was invalid? A widely held view is that a non-local theory is needed—another is that hidden variable theory should be contextual. In this article, derivations of the CHSH Bell inequality based on various combinations of locality/non-locality and non-contextuality/contextuality assumptions are outlined, leading to the conclusion that for a hidden variable theory to underpin quantum theory, it must be both non-local and contextual. Non-locality or contextuality alone is insufficient. However, it is also concluded that the realism assumption is questionable. At present an alternative approach of augmenting quantum theory by adding additional features—such as the deterministic trajectories of Bohmian mechanics (also a non-local theory) may be a better route for completeness.
Abstract Quantum theory has accounted for a wide range of physical effects in systems ranging from microscopic to macroscopic. However, it is associated with various paradoxes—such as the well-known EPR example. Einstein’s suggestion that the theory was incomplete has led to numerous attempts to underpin quantum theory by a hidden variable theory, including the Bell approach. This approach was originally based on certain assumptions: (i) realism (ii) locality and (iii) non-contextuality. In its developed form, the theory involved only probabilities for outcome values of system observables. For systems involving two sub-systems, the hidden variable theory predicted that correlations between measurement outcomes of observable pairs of from different sub-systems would satisfy certain—such as the CHSH Bell inequality. The inequality was inconsistent with quantum theory predictions and with experiment. Hence, the question arose: Which of the three assumptions was invalid? A widely held view is that a non-local theory is needed—another is that hidden variable theory should be contextual. In this article, derivations of the CHSH Bell inequality based on various combinations of locality/non-locality and non-contextuality/contextuality assumptions are outlined, leading to the conclusion that for a hidden variable theory to underpin quantum theory, it must be both non-local and contextual. Non-locality or contextuality alone is insufficient. However, it is also concluded that the realism assumption is questionable. At present an alternative approach of augmenting quantum theory by adding additional features—such as the deterministic trajectories of Bohmian mechanics (also a non-local theory) may be a better route for completeness.
Abstract In this paper we consider the description by a general Bell-type non-local hidden variable theory of bipartite quantum states with two observables per sub-system. We derive Bell inequalities of the Collins-Gisin.-Liden-Massar-Popescu type which involve combinations of the probabilities of related outcomes for measurements for the four pairs of sub-system observables. It is shown that the corresponding quantum theory expressions violate the Bell inequalities in the case of the maximally entangled state of the bipartitite system. The CHSH Bell inequality is also derived from this general CGLMP Bell-type non-local hidden variable theory. This shows that quantum theory can not be underpinned by a Bell-type non-local hidden variable theory. So as a general Bell-type local hidden variable theory has already been shown to conflict with quantum theory, it follows that quantum theory can not be understood in terms of any CGLMP Bell-type hidden variable theory—local or non-local.
Several decades of theory and experiment into EPR correlations have led to the widely held belief that reality is non-local, in spite of the fact that this violates special relativity. To date, no experiment has shown a violation of special relativity, and EPR experiments do not demonstrate the existence of superluminal information exchange, merely correlations which violate certain inequalities. Every “loophole” in these hidden variable theories has been thought plugged. However, there is much confusion in the literature due to conflation of the terms ‘locality”, “realism”, “hidden variables”, “non-contextuality”. The presence of local hidden variables is thought to necessarily lead to a Kolmogorov probability structure (hence non-contextuality), but this is an assumption, one which is not true in general once context effects are taken into account. Treated as an observational theory, several authors have shown no incompatibility between quantum mechanics and locality, and that the Bell scenario is actually about whether reality is contextual. This paper proposes a descriptive theory by assuming a generated reality (following Whitehead’s Process Theory) which can violate the principle of continuity and possess non-Kolmogorov probability structure, and reproduce the results of non-relativistic quantum mechanics, while allowing only causally local information exchange, without hidden variables. A generated reality is thus compatible with both quantum mechanics and special relativity, reproducing all of the results expected from quantum mechanics while still maintaining causally local realism. This process model thus appears to be an ideal candidate for developing theories for the unification of quantum mechanics and general relativity.
Bell’s inequality is an empirical constraint on theories with hidden variables, which Einstein, Podolsky and Rosen argued are needed to explain observed perfect correlations if keeping locality. One way to deal with the empirical violation of Bell’s inequality is by openly embracing nonlocality, in a theory like the pilot-wave theory. Nonetheless, recent proposals have revived the possibility that one can avoid nonlocality by resorting to superdeterministic theories. These are local hidden variables theories which violate statistical independence which is one assumption of Bell’s inequality. In this paper I compare and contrast these two hidden variable strategies: the pilot-wave theory and superdeterminism. I show that even if the former is nonlocal and the other is not, both are contextual. Nonetheless, in contrast with the pilot-wave theory, superdeterminist contextuality makes it impossible to test the theory (which therefore becomes unfalsifiable and unconfirmable) and renders the theory uninformative (measurement results tell us nothing about the system). It is questionable therefore whether a theory with these features is worth its costs.
Quantum non-locality tests have been of interest since the original EPR paper. The present paper discusses whether the CGLMP (Bell) inequalities obtained by Collins et al are possible tests for showing that quantum theory is not underpinned by local hidden variable theory (LHVT). It is found by applying Fine's theorem that the CGLMP approach involves a LHVT for the probabilities associated with measurements on two observables (each from one of the two sub-subsystems), even though the underlying probabilities for measurements of all four observables involve a hidden variable theory which is not required to be local. Although the CGLMP inequalities involve probabilities for measurements of one observable per sub-system and are compatible with the Heisenberg uncertainty principle, there is no unambiguous quantum measurement process linked to the probabilities in the CGLMP inequalities. Quantum measurements corresponding to the different classical measurements that give the same CGLMP probability are found to yield different CGLMP probabilities. However, violation of a CGLMP inequality based on any one of the possible quantum measurement sequences is sufficient to show that the Collins et al LHVT does not predict the same results as quantum theory. This is found to occur for a state considered in their paper - though for observables whose physical interpretation is unclear. In spite of the problems of comparing the HVT inequalities with quantum expressions, it is concluded that the CGLMP inequalities are indeed suitable for ruling out local hidden variable theories. The state involved could apply to a macroscopic system, so the CGLMP Bell inequalities are important for finding cases of macroscopic violations of Bell locality. Possible experiments in double-well Bose condensates involving atoms with two hyperfine components are discussed.
I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. I also identify similar oversight in the GHZ variant of Bell's theorem, invalidating its claim of having found an inconsistency in the premisses of the argument by EPR for completing quantum mechanics. Conceptually, the oversight in both Bell's theorem and its GHZ variant traces back to the oversight in von~Neumann's theorem against hidden variable theories identified by Grete Hermann in the 1930s.
It is known that "quantum non locality", leading to the violation of Bell's inequality and more generally of classical local realism, can be attributed to the conjunction of two properties, which we call here elementary locality and predictive completeness. Taking this point of view, we show again that quantum mechanics violates predictive completeness, allowing the making of contextual inferences, which can, in turn, explain why quantum non locality does not contradict relativistic causality. An important question remains: if the usual quantum state ψ is predictively incomplete, how do we complete it? We give here a set of new arguments to show that ψ should be completed indeed, not by looking for any "hidden variables", but rather by specifying the measurement context, which is required to define actual probabilities over a set of mutually exclusive physical events.