Hidden variable models for quantum theory cannot have any local part

It was shown by Bell that no local hidden variable model is compatible with quantum mechanics. If, instead, one permits the hidden variables to be entirely nonlocal, then any quantum mechanical predictions can be recovered. In this Letter, we consider general hidden variable models which can have both local and nonlocal parts. We show the existence of (experimentally verifiable) quantum correlations that are incompatible with any hidden variable model having a nontrivial local part, such as the model proposed by Leggett.     

Is the Devil in h?

This note is a part of my effort to rid quantum mechanics (QM) nonlocality. Quantum nonlocality is a two faced Janus: one face is a genuine quantum mechanical nonlocality (defined by the Lüders’ projection postulate). Another face is the nonlocality of the hidden variables model that was invented by Bell. This paper is devoted the deconstruction of the latter. The main casualty of Bell’s model is that it straightforwardly contradicts Heisenberg’s uncertainty and Bohr’s complementarity principles generally. Thus, we do not criticize the derivation or interpretation of the Bell inequality (as was done by numerous authors). Our critique is directed against the model as such. The original Einstein-Podolsky-Rosen (EPR) argument assumed the Heisenberg’s principle without questioning its validity. Hence, the arguments of EPR and Bell differ crucially, and it is necessary to establish the physical ground of the aforementioned principles. This is the quantum postulate: the existence of an indivisible quantum of action given by the Planck constant. Bell’s approach with hidden variables implicitly implies rejection of the quantum postulate, since the latter is the basis of the reference principles.     

Closing the detection loophole in the Bell test for local models based upon three hidden variables

Locality and fair sampling are proved to be contradictory assumptions in hidden variable models of the Bell test that are based upon a 3-dimensional sample space. This result makes the class of 3-dimensional hidden variable models incompatible with quantum mechanics in the ideal case, independently of detection efficiencies.     

Necessary and sufficient detection efficiency for the mermin inequalities

We prove that the threshold detection efficiency for a loophole-free Bell experiment using an n-qubit Greenberger-Horne-Zeilinger state and the correlations appearing in the n-partite Mermin inequality is n/(2n-2). If the detection efficiency is equal to or lower than this value, there are local hidden variable models that can simulate all the quantum predictions. If the detection efficiency is above this value, there is no local hidden variable model that can simulate all the quantum predictions.     

When is a hidden variable theory compatible with quantum mechanics?

This paper is devoted to a study of some of the basic conditions which have to be satisfied by a hidden variable theory in order that it can reproduce the quantum mechanical probabilities. Of course one such condition, which emerges from the important theorem of Bell, is that a hidden variable theory has to be non-local. It is shown that a hidden variable theory is also incompatible with the conventional interpretation of mixed states and the mixing operation in quantum theory. It is therefore concluded that, apart from being non-local, a hidden variable theory would also necessarily violate the usual assumption of quantum theory that the density operator provides an adequate characterization of any ensemble of systems, pure or mixed.     

POVMs and hidden variables

Recent results by Paul Busch and Adán Cabello claim to show that by appealing to POVMs, non-contextual hidden variables can be ruled out in two dimensions. While the results of Busch and Cabello are mathematically correct, interpretive problems render them problematic as no hidden variable proofs.     

Comment on “A simple experiment to test Bell’s inequality”, J.-M. Vigoureux

In the above paper, it is claimed that with a particular use of the Bell inequality, a simple single photon experiment could be performed to show the impossibility of any deterministic hidden variable theory in quantum optics. A careful analysis of the concept of probability for hidden variables and a detailed discussion of the hidden variable model of de Broglie–Bohm show that the reasoning and main conclusion of this paper are not correct.     

On the implication of Bell’s probability distribution and proposed experiments of quantum measurement

In derivating of Bell’s inequalities, the probability distribution is supposed to be a function only of a hidden variable. We point out that the true implication of the probability distribution of Bell’s correlation function is the distribution of joint measurement outcomes on the two sides. It is therefore a function of both the hidden variable and the settings. In this case, Bell’s inequalities fail. Our further analysis shows that Bell’s locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, and hence violation of Bell’s inequalities cannot rule out the existence of local hidden variable. To explain the results of EPR-type experiments, we suppose that a polarization-entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any nonlocal influence on it. Then several experiments are suggested to reveal the components of the polarization-entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with the same quantum state behave the same under the same measuring conditions.     

Symmetric extensions of quantum States and local hidden variable theories

While all bipartite pure entangled states violate some Bell inequality, the relationship between entanglement and nonlocality for mixed quantum states is not well understood. We introduce a simple and efficient algorithmic approach for the problem of constructing local hidden variable theories for quantum states. The method is based on constructing a so-called symmetric quasiextension of the quantum state that gives rise to a local hidden variable model with a certain number of settings for the observers Alice and Bob.     

The Modified Bell Inequality and Its Physical Implications in the ESR Model

The extended semantic realism (ESR) model proposes a theoretical perspective which reinterprets quantum probabilities as conditional on detection rather than absolute and embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual, hence local, framework. The assumptions needed to prove the Bell inequality therefore hold in the ESR model, but we show that the Bell inequality must be substituted in it by the modified Bell inequality and that the standard quantum expectation values, when reinterpreted as proposed by the ESR model, do not violate the latter inequality. Hence the long-standing conflict between ??local realism?? and QM is settled in the ESR model. Finally we provide an elementary example of a prediction that might be used to check whether the ESR model is correct.     

Von Neumann’s ‘No Hidden Variables’ Proof: A Re-Appraisal

Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann’s ‘no hidden variables’ proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense ‘like Bohm’s theory.’ His seminal result provides a positive answer to the question. I argue that Bell’s analysis misconstrues von Neumann’s argument. What von Neumann proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the ‘beables’ of the theory, to use Bell’s term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm’s theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system.     

Hidden variables and locality

Bell's problem of the possibility of a local hidden variable theory of quantum phenomena is considered in the context of the general problem of representing the statistical states of a quantum mechanical system by measures on a classical probability space, and Bell's result is presented as a generalization of Maczynski's theorem for maximal magnitudes. The proof of this generalization is shown to depend on the impossibility of recovering the quantum statistics for sequential probabilities in a classical representation without introducing a randomization process for the hidden variables. Hidden variable theories that exclude such a randomization process are termed strict, and it is shown that the class of local hidden variable theories is included in the class of strict theories. A counterargument by Freedman and Wigner is evaluated with reference to Clauser's extension of a hidden variable model proposed by Bell.     

On Noncontextual,Non-Kolmogorovian Hidden Variable Theories

One implication of Bell’s theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig (Br J Philos Sci 66(4): 905–927, 2015) to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities.     

Research on hidden variable theories: A review of recent progresses

Quantum Mechanics (QM) is one of the pillars of modern physics: an impressive amount of experiments have confirmed this theory and many technological applications are based on it. Nevertheless, at one century since its development, various aspects concerning its very foundations still remain to be clarified. Among them, the transition from a microscopic probabilistic world into a macroscopic deterministic one and quantum non-locality. A possible way out from these problems would be if QM represents a statistical approximation of an unknown deterministic theory.This review is addressed to present the most recent progresses on the studies related to hidden variable theories (HVT), both from an experimental and a theoretical point of view, giving a larger emphasis to results with a direct experimental application.More in details, the first part of the review is a historical introduction to this problem. The Einstein–Podolsky–Rosen argument and the first discussions about HVT are introduced, describing the fundamental Bell's proposal for a general experimental test of every local HVT and the first attempts to realise it.The second part of the review is devoted to elucidate the recent progresses toward a conclusive Bell inequalities experiment obtained with entangled photons and other physical systems.Finally, the last sections are targeted to shortly discuss non-local HVT.     

Separability of Quantum States vs. Original Bell (1964) Inequalities

All separable states satisfy all Bell-type inequalities, which involve as their assumption only existence of local realistic (local hidden variable) models of the correlations of spatially separated systems, observed by two or more observers making independent decisions on what to measure (free will). The recent observation by Loubenets, that some separable states do not satisfy the original Bell inequality (1964) has no consequences whatsoever for the studies of the relation of separability with local realism. The original Bell inequality was derived using an additional assumption that the local results for a certain pair of local settings reveal perfect Einstein–Podolsky–Rosen (EPR) correlations. Therefore violation of this inequality by some quantum predictions implies that either (i) the predictions do not allow a local realistic model, or (ii) the predictions do not have the required EPR correlations, or finally both (i) and (ii).     

Constraints on Determinism: Bell Versus Conway–Kochen

Bell’s Theorem from Physics 36:1–28 (1964) and the (Strong) Free Will Theorem of Conway and Kochen from Notices AMS 56:226–232 (2009) both exclude deterministic hidden variable theories (or, in modern parlance, ‘ontological models’) that are compatible with some small fragment of quantum mechanics, admit ‘free’ settings of the archetypal Alice and Bob experiment, and satisfy a locality condition akin to parameter independence. We clarify the relationship between these theorems by giving reformulations of both that exactly pinpoint their resemblance and their differences. Our reformulation imposes determinism in what we see as the only consistent way, in which the ‘ontological state’ initially determines both the settings and the outcome of the experiment. The usual status of the settings as ‘free’ parameters is subsequently recovered from independence assumptions on the pertinent (random) variables. Our reformulation also clarifies the role of the settings in Bell’s later generalization of his theorem to stochastic hidden variable theories.     

A Kochen-Specker inequality from a SIC

Yu and Oh (eprint) [1] have given a state-independent proof of the Kochen-Specker theorem in three dimensions using only 13 rays. The proof consists of showing that a non-contextual hidden variable theory necessarily leads to an inequality that is violated by quantum mechanics. We give a similar proof making use of 21 rays that constitute a SIC (symmetric informationally-complete positive operator-valued measure) and a complete set of MUB (mutually unbiased bases). A theory-independent inequality is also presented using the same 21 rays, as required for experimental tests of contextuality.     

Strengthened Bell inequalities for orthogonal spin directions

We provide bounds on correlations of locally orthogonal observables in two-qubit separable states. These bounds strengthen the Bell inequality and improve upon some alternative entanglement criteria. They provide necessary and sufficient criteria for separability of pure states and test the correlations allowed by local hidden variable models against those allowed by separable quantum states.