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.     

What is a hidden variable theory of quantum phenomena?

The purpose of this paper is to clarify the relationship between existing so-called hidden variable theories of quantum phenomena and some well-known proofs, such as those of von Neumann, Jauch and Piron, and Kochen and Specker, which purport to establish that no such theory is possible. The proof of Kochen and Specker, which is a stronger version of von Neumann's result, demonstrates the impossibility of embedding the algebraic structure of physical parameters of the quantum theory, represented by the self-adjoint Hubert space operators, into the commutative algebra of real-valued functions on a phase space of hidden states. This is a necessary condition for a hidden variable extension of the quantum theory in the usual sense of a statistical mechanical derivation of the statistical theorems of the quantum theory in the classical manner. No existing so-called hidden variable theory is a counter-example to von Neumann's proof. The early 1951 hidden variable theory of Bohm and the recent theory of Bohm and Bub are not in fact hidden variable theories in the usual sense of the term. Since the term hidden variable theory is justifiably used to denote the kind of theory rejected by von Neumann, Jauch and Piron, and Kochen and Specker, it is suggested that the term should not be used as a label for the theories considered by Bohm and other workers in this field. Such theories could be regarded as fundamentally compatible with the original Copenhagen interpretation of the quantum theory, as expressed by Bohr.Supported by the National Science Foundation.     

Proof of a quantum mechanical nonlocal influence

First it is proved that, in a deterministic theory, Malus' law requires that, if a photon is successively transmitted by two polarizers with appropriately chosen settings, the first transmission influences a hidden variable (co-) determining the second one. We derive from this that in an ideal EPR experiment (giving the result predicted by quantum mechanics for two correlated photons transmitted by two polarizers with suitably chosen settings) there has to be a nonlocal influence from the first transmission interaction to the second. Subsequently we argue that we can abandon determinism as an assumption so that the locality hypothesis is in any case untenable if the predictions of quantum mechanics are all correct.     

On the possibility of a phase-space reconstruction of quantum statistics: A refutation of the Bell-Wigner locality argument

J. S. Bell's argument that only nonlocal hidden variable theories can reproduce the quantum statistical correlations of the singlet spin state in the case of two separated spin-1/2 particles is examined in terms of Wigner's formulation. It is shown that a similar argument applies to a single spin-1/2 particle, and that the exclusion of hidden variables depends on an obviously untenable assumption concerning conditional probabilities. The problem of completeness is discussed briefly, and the grounds for rejecting a phase-space reconstruction of the quantum statistics are clarified.Research supported by the National Science Foundation.     

Niels Bohr’s Generalization of Classical Mechanics

We clarify Bohrs interpretation of quantum mechanics by demonstrating the central role played by his thesis that quantum theory is a rational generalization of classical mechanics. This thesis is essential for an adequate understanding of his insistence on the indispensability of classical concepts, his account of how the quantum formalism gets its meaning, and his belief that hidden variable interpretations are impossible.     

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.     

A Model with Quantum Logic, but Non-Quantum Probability: The Product Test Issue

We introduce a model with a set of experiments of which the probabilities of the outcomes coincide with the quantum probabilities for the spin measurements of a quantum spin- particle. Product tests are defined which allow simultaneous measurements of incompatible observables, which leads to a discussion of the validity of the meet of two propositions as the algebraic model for conjunction in quantum logic. Although the entity possesses the same structure for the logic of its experimental propositions as a genuine spin- quantum entity, the probability measure corresponding with the meet of propositions using the Hilbert space representation and quantum rules does not render the probability of the conjunction of the two propositions. Accordingly, some fundamental concepts of quantum logic, Piron-products, classical systems and the general problem of hidden variable theories for quantum theory are discussed.     

Interferometric Computation Beyond Quantum Theory

There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer. There has been recent foundational interest in theories beyond quantum theory. Here, we present a generalized formulation of computation in the context of a many-armed interferometer, and explore how theories can differ from quantum theory and still perform distributed calculations in this set-up. We shall see that quaternionic quantum theory proves a suitable candidate, whereas box-world does not. We also find that a classical hidden variable model first presented by Spekkens (Phys Rev A 75(3): 32100, 2007) can also be used for this type of computation due to the epistemic restriction placed on the hidden variable.     

Isospin as a hidden variable

A hidden isospin variable is coupled to the spin of particles observed in an EPR experiment. For spin-1/2 it is shown that isospin i3/2 is sufficient to ensure a locally realistic spin distribution. For spin-1, examples of violation of the Mermin-Schwarz inequalities in the case of i=0 are shown satisfied with isospin. The general feature of a softening of quantum nonlocality with isospin is suggested, as well as applications to quantum physics at high energy.     

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.     

Bell's theorem,inference, and quantum transactions

Bell's theorem is expounded as an analysis in Bayesian inference. Assuming the result of a spin measurement on a particle is governed by a causal variable internal (hidden, local) to the particle, one learns about it by making a spin measurement; thence about the internal variable of a second particle correlated with the first; and from there predicts the probabilistic result of spin measurements on the second particle. Such predictions are violated by experiment: locality/causality fails. The statistical nature of the observations rules out signalling; acausal, superluminal, or otherwise. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of experiment imply that it has a nonlocal/acausal interpretation. Cramer's newtransactional interpretation, which incorporates this feature by adapting the Wheeler-Feynman idea of advanced and retarded processes to the quantum laws, is advocated. It leads to an invaluable way of envisaging quantum processes. The usual paradoxes melt before this, and one, the delayed choice experiment, is chosen for detailed inspection. Nonlocality implies practical difficulties in influencing hidden variables, which provides a very plausible explanation for why they have not yet been found; from this standpoint, Bell's theoremreinforces arguments in favor of hidden variables.     

Quantum adiabatic approximation,quantum action,and Berry's phase

An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry's adiabatic phase and its semi-classical limit are also discussed.     

Approximate Hidden Variables

The usual definition of (non-contextual) hidden variables is found to be too restrictive, in the sense that, according to it, even some classical systems do not admit hidden variables. A more general concept is introduced and the term approximate hidden variables is used for it. This new concept avoids the aforementioned problems, since all classical systems admit approximate hidden variables. Standard quantum systems do not admit approximate hidden variables, unless the corresponding Hilbert space is 2-dimensional. However, an appropriate non-standard quantum system, which arises by focussing on momentum and position and neglecting the remaining observables, admits approximate hidden variables. Systems associated with JBW-algebras (resp. von Neumann algebras) and satisfying some mild conditions admit approximate hidden variables iff they are classical, that is, iff the corresponding JBW-algebra (resp. von Neumann algebra) is associative (resp. commutative).     

A Simple Model for an Objective Interpretation of Quantum Mechanics

An SR model is presented that shows how an objective (noncontextual and local) interpretation of quantum mechanics can be constructed, which contradicts some well-established beliefs following from the standard interpretation of the theory and from known no-go theorems. The SR model is not a hidden variables theory in the standard sense, but it can be considered a hidden parameters theory which satisfies constraints that are weaker than those usually imposed on standard hidden variables theories. The SR model is also extended in a natural way that shows how a broader theory embodying quantum mechanics can be envisaged which is realistic in a semantic sense, hence compatible with various realistic perspectives.     

The incompatibility between local hidden variable theories and the fundamental conservation laws

I discuss in detail the result that the Bell’s inequalities derived in the context of local hidden variable theories for discrete quantized observables can be satisfied only if a fundamental conservation law is violated on the average. This result shows that such theories are physically nonviable, and makes the demarcating criteria of the Bell’s inequalities redundant. I show that a unique correlation function can be derived from the validity of the conservation law alone and this coincides with the quantum mechanical correlation function. Thus, any theory with a different correlation function, like any local hidden variable theory, is incompatible with the fundamental conservation laws and space-time symmetries. The results are discussed in the context of two-particle singlet and triplet states, GHZ states, and two-particle double slit interferometry. Some observations on quantum entropy, entanglement, and nonlocality are also discussed.     

Quantum Versus Stochastic Processes and the Role of Complex Numbers

We argue that the complex numbers are an irreducible object of quantum probability: this can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having the complex phases as primitive ingredient implies that we need to accept nonadditive probabilities. This has the desirable consequence of removing constraints of standard theorems about the possibility of describing quantum theory with commutative variables. Motivated by the formalism of consistent histories and keeping an analogy with the theory of stochastic processes, we develop a (statistical) theory of quantum processes: they are characterized by the introduction of a density matrix on phase space paths (it thus includes phase information) and fully reproduces quantum mechanical predictions. We can write quantum differential equations (in analogy to Langevin equation) that could be interpreted as referring to individual quantum systems. We describe the reconstruction theorem by which a quantum process can yield the standard Hilbert space structure if the Markov property is imposed. We discuss the relevance of our results for the interpretation of quantum theory (a sample space is possible if probabilities are nonadditive) and quantum gravity (the Hilbert space arises here after the consideration of a background causal structure).     

Application of the 1H(d,2He)n reaction to the EPR paradox

The test of the local hidden variable theories involving strongly interacting pairs of massive spin- $\frac{1}{2}$ hadrons, in a nearly loophole-free experimental setting is reported. The spin correlation function is deduced to be C _exp(π/4) = ?0.73 ± 0.08. This result is in agreement with non-local quantum mechanical prediction and it violates the Bell’s inequality of C _Bell(π/4) ≥ ?0.5 at a confidence level of 99.3%.     

Algebraic constraints on hidden variables

In the contemporary discussion of hidden variable interpretations of quantum mechanics, much attention has been paid to the no hidden variable proof contained in an important paper of Kochen and Specker. It is a little noticed fact that Bell published a proof of the same result the preceding year, in his well-known 1966 article, where it is modestly described as a corollary to Gleason's theorem. We want to bring out the great simplicity of Bell's formulation of this result and to show how it can be extended in certain respects.Work on this paper was partially supported by National Science Foundation Grants SOC 76-82113 and SOC 76-10659.     

Hidden Measurements,Hidden Variables and the Volume Representation of Transition Probabilities

We construct, for any finite dimension n, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For n=2 our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions n . 3 We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our hidden variables scheme and their relationship with the results of Fine [J. Math. Phys. 23 1306 (1982)].