Bell inequalities have long served as the standard criterion to verify local hidden variable theories. Experimental violations of these inequalities are commonly interpreted as fundamental quantum nonlocality at the microscopic scale. This paper proves rigorously that Bell inequalities hold only under three prerequisites: countable discrete variables, countably additive probability measures, and classical absolute simultaneity. The state space of entangled quantum systems forms an uncountable infinite continuum. By the property of Vitali non-measurable sets in measure theory, the integral deduction of the CHSH inequality collapses naturally over continuous state spaces. Furthermore, special relativity excludes universal global time synchronization, so the absolute time embedded in Bell’s theory is an artificial assumption inconsistent with physical reality. Inspired by the framework dependence of probability revealed in Bertrand’s paradox, we demonstrate that the excess correlation observed in experiments is an apparent effect caused by the mismatch between classical measurement frameworks and the intrinsic continuum property of quantum systems, rather than evidence against local hidden variables. Statistical correlation between entangled particles originates from their shared evolutionary sequence defined at pair generation, with no superluminal interaction, fully complying with relativistic causality. This paper does not explore the ultimate origin of entanglement, but treats it as a given primitive physical law. Two falsifiable predictions are proposed to distinguish the continuum model from standard quantum mechanics. The conclusion is that hidden variable theories have never been falsified by experiments; they only need to be extended from discrete forms to continuum models compatible with relativity.
Bell inequalities have long served as the standard criterion to verify local hidden variable theories. Experimental violations of these inequalities are commonly interpreted as fundamental quantum nonlocality at the microscopic scale. This paper proves rigorously that Bell inequalities hold only under three prerequisites: countable discrete variables, countably additive probability measures, and classical absolute simultaneity. The state space of entangled quantum systems forms an uncountable infinite continuum. By the property of Vitali non-measurable sets in measure theory, the integral deduction of the CHSH inequality collapses naturally over continuous state spaces. Furthermore, special relativity excludes universal global time synchronization, so the absolute time embedded in Bell’s theory is an artificial assumption inconsistent with physical reality. Inspired by the framework dependence of probability revealed in Bertrand’s paradox, we demonstrate that the excess correlation observed in experiments is an apparent effect caused by the mismatch between classical measurement frameworks and the intrinsic continuum property of quantum systems, rather than evidence against local hidden variables. Statistical correlation between entangled particles originates from their shared evolutionary sequence defined at pair generation, with no superluminal interaction, fully complying with relativistic causality. This paper does not explore the ultimate origin of entanglement, but treats it as a given primitive physical law. Two falsifiable predictions are proposed to distinguish the continuum model from standard quantum mechanics. The conclusion is that hidden variable theories have never been falsified by experiments; they only need to be extended from discrete forms to continuum models compatible with relativity.
Bell–CHSH inequalities are commonly interpreted as demonstrating the incompatibility between quantum mechanics and local hidden-variable theories based on pre-existing measurement outcomes. In standard derivations of the CHSH inequality, however, it is implicitly assumed that all possible measurement settings can be assigned simultaneous values within a single global hidden-variable structure. In this paper, we reconsider the Bell–CHSH framework from the viewpoint of the Intrinsic Observability Framework (IOF). We argue that the central issue is not the experimental violation of the inequality itself, but the applicability of the global value assignment assumed in its derivation. In particular, the simultaneous definition of outcomes associated with incompatible measurement settings presupposes a common operational structure shared across all observational configurations. Within IOF, observable outcomes are not treated as globally pre-existing values independent of observational accessibility and decomposition structure. Instead, measurement outcomes are associated with operationally accessible configurations determined through observable–hidden relations. From this perspective, the global assignment required for the CHSH derivation is not operationally justified. We further show that this viewpoint remains fully consistent with the standard quantum-mechanical correlations observed in Bell-type experiments. The present analysis suggests that the Bell–CHSH inequality may be interpreted not as a direct contradiction between quantum mechanics and locality itself, but rather as a limitation of globally defined pre-existing value assignments across incompatible observational configurations.
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.
Eighty years ago Einstein, Podolsky and Rosen demonstrated that instantaneous reduction of wave function, believed to describe completely a pair of entangled physical systems, led to EPR paradox. The paradox disappears in statistical interpretation of quantum mechanics (QM) according to which a wave function describes only an ensemble of identically prepared physical systems. QM predicts strong correlations between outcomes of measurements performed on different members of EPR pairs in far-away locations. Searching for an intuitive explanation of these correlations John Bell analysed so called local realistic hidden variable models and proved that correlations consistent with these models satisfy Bell inequalities which are violated by some predictions of QM and by experimental data. Several different local models were constructed and inequalities proven. Some eminent physicists concluded that Nature is definitely nonlocal and that it is acting according to a law of nonlocal randomness. According to these law perfectly random, but strongly correlated events, can be produced at the same time in far away locations and a local and causal explanation of their occurrence cannot be given. We strongly disagree with this conclusion and we prove the contrary by analysing in detail some influential finite sample proofs of Bell and CHSH inequalities and so called Quantum Randi Challenges. We also show how one can win so called Bell's game without violating locality of Nature. Nonlocal randomness is inconsistent with local quantum field theory, with standard model in elementary particle physics and with causal laws and adaptive dynamics prevailing in the surrounding us world. The experimental violation of Bell-type inequalities does not prove the nonlocality of Nature but it only confirms a contextual character of quantum observables and gives a strong argument against counterfactual definiteness and against a point of view according to which experimental outcomes are produced in irreducible random way.
Abstract Quantum mechanics predicts faster-than-light information transportation in order to satisfy quantum entanglement experimental results. Such a prediction violates the locality principle and Bell’s inequality. Quantum mechanisms also state that the world is intrinsically probabilistic. Such an idea was not welcome by many scientists including Einstein, Podolsky and Rosen. To resolve such a dilemma, the ”Many-world” theory was presented by Hugh Everett in 1975. Such a theory asserts that the universal wave function is real and that there is no wave function collapse. Also, it states that all possible outcomes of quantum measurements are physically realized in other parallel universes. Besides, in quantum mechanics, the observer effect defines a state transition of quantum objects. Such a phenomenon affirms that quantum objects behave differently depending on being observed or not. In this paper, we present few assumptions based on which we build a classical hidden-variable-based system that can violate Bell’s inequality. Therefore, adopting similar assumptions in explaining quantum entanglement could lead to a better explanation of quantum mechanics without violating the locality principle or adopting such weird assumptions. Hence, one can conclude that extending Quantum Mechanism by a set of similar hidden variables can resolve such a paradox and predict the quantum entanglement experimental results without violating the locality principle.
It is one of the most remarkable features of quantum physics that measurements on spatially separated systems cannot always be described by a locally causal theory. In such a theory, the outcomes of local measurements are determined in advance solely by some unknown (or hidden) variables and the choice of local measurements. Correlations that are allowed within the framework of a locally causal theory are termed classical. Typically, the fact that quantum mechanics does not always result in classical correlations is revealed by the violation of Bell inequalities, which are constraints that have to be satisfied by any classical correlations. It has been known for a long time that entanglement is necessary to demonstrate nonclassical correlations, and hence a Bell inequality violation. However, since some entangled quantum states are known to admit explicit locally causal models, the exact role of entanglement in Bell inequality violation has remained obscure. This thesis provides both a comprehensive review on these issues as well as a report on new discoveries made to clarify the relationship between entanglement and Bell inequality violation. In particular, within the framework of a standard Bell experiment, i.e., a Bell inequality test that is directly performed on a single copy of a quantum state r, we have derived two algorithms to determine, respectively, a lower bound and an upper bound on the strength of correlations that r can offer for any given Bell inequality. Both of these algorithms make use convex optimization techniques in the form of a semidefinite program. By examples, we show that these algorithms can often be used in tandem, in conjunction with convexity arguments, to determine if a quantum state can offer nonclassical correlations and hence violates a given Bell inequality. On the other hand, since a standard Bell experiment typically involves measurements over many copies of the quantum systems, we have also investigated the possibility of enhancing the strength of nonclassical correlation by, instead, performing collective measurements on multiple copies of the quantum systems. Our findings show that even without postselection, such joint measurements may also lead to stronger nonclassical correlations, and hence a better Bell inequality violation. Meanwhile, previous studies have indicated that entangled state admitting locally causal models may still lead to observable nonclassical correlations if, prior to a standard Bell experiment, the state is subjected to some appropriate local preprocessing. This phenomenon of hidden nonlocality was discovered more than a decade ago, but to date, it is still not known if all entangled states can demonstrate nonclassical correlations through these more sophisticated Bell experiments. A key result in this thesis then consists of showing that for all bipartite entangled states, observable nonclassical correlations, in the form of a Bell-CHSH inequality violation, can indeed be derived if we allow both local preprocessing and the usage of shared ancillary state which by itself does not violate the Bell-CHSH inequality. This establishes a kind of equivalence between bipartite entanglement and states that cannot be simulated by classical correlations. In summary, for a standard Bell experiment where no local preprocessing on a quantum state r is allowed, we have provided two algorithms that can be used in tandem to determine if r can be simulated by a locally causal theory, whereas in the scenario where local preprocessing is allowed, we have demonstrated that bipartite entangled states are precisely those which cannot always be simulated classically.
The Bell inequality is thought to be a common constraint shared by all models of local hidden variables that aim to describe the entangled states of two qubits. Since the inequality is violated by the quantum mechanical description of these states, it purportedly allows distinguishing in an experimentally testable way the predictions of quantum mechanics from those of models of local hidden variables and, ultimately, ruling the latter out. In this paper, we show, however, that the models of local hidden variables constrained by the Bell inequality all share a subtle, though crucial, feature that is not required by fundamental physical principles and, hence, it might not be fulfilled in the actual experimental setup that tests the inequality. Indeed, the disputed feature neither can be properly implemented within the standard framework of quantum mechanics and it is even at odds with the fundamental principle of relativity. Namely, the proof of the inequality requires the existence of a preferred absolute frame of reference (supposedly provided by the lab) with respect to which the hidden properties of the entangled particles and the orientations of each one of the measurement devices that test them can be independently defined through a long sequence of realizations of the experiment. We notice, however, that while the relative orientation between the two measurement devices is a properly defined physical magnitude in every single realization of the experiment, their global rigid orientation with respect to a lab frame is a spurious gauge degree of freedom. Following this observation, we were able to explicitly build a model of local hidden variables that does not share the disputed feature and, hence, it is able to reproduce the predictions of quantum mechanics for the entangled states of two qubits.