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arXiv:0907.2396

A counter-argument to "Hidden variable models for quantum theory cannot have any local part" arXiv:0801.2218

Sofia Wechsler

Published Jul 14, 2009• 1 citation
arXivHTMLPDF

Abstract

Colbeck and Renner [arXiv:0801.2218] analyzed a class of combined models for entanglements in which local and non-local hidden variables cooperate for producing the measurement results. They came to the conclusion that the measurement results are fully independent of the local components of the hidden variables. Their conclusion is based mainly on an assumption on the local hidden variables, assumption similar to the non-signaling property of probabilities of observables' values. In the present text it is proved that hidden variables are not observables, so their distributions of probabilities do not necessarily possess the non-signaling property. Also, a counter-example is brought to the Colbeck and Renner assumption, showing that their type of models and conclusion are not general. The question whether hidden variables, local or non-local, exist or not, remains open.

DOI
10.48550/arxiv.0907.2396

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Colbeck and Renner [arXiv:0801.2218] analyzed a class of combined models for entanglements in which local and non-local hidden variables cooperate for producing the measurement results. They came to the conclusion that the measurement results are fully independent of the local components of the hidden variables. Their conclusion is based mainly on an assumption on the local hidden variables, assumption similar to the non-signaling property of probabilities of observables' values. In the present text it is proved that hidden variables are not observables, so their distributions of probabilities do not necessarily possess the non-signaling property. Also, a counter-example is brought to the Colbeck and Renner assumption, showing that their type of models and conclusion are not general. The question whether hidden variables, local or non-local, exist or not, remains open.
A recent article of Colbeck and Renner tackled the problem whether entanglements may be explained by combined models of local and non-local hidden variables. To the difference from previous works they considered models in which each pair of entangled particles behaves in the same way, and the particles in the pair are equivalent, i.e. each of them produces its response to a measurement according to both local and non-local hidden variables. Their article aimed at proving that the local hidden variable component in such models has no effect on the measurement results, i.e. only the non-local variables are relevant. However, their proof deals with a very restrictive case and assumes questionable constraints on the hidden variables. The present text studies the Colbeck and Renner class of models on a less restrictive case and under no constraints on the hidden variables. It is shown again that the local component cannot have any influence on the results. However, the Colbeck and Renner class of models is not the only one possible. A different class is described, and it admits local hidden variables by the side of the non-local influence. This class presents a couple of advantages.
We criticize Colbeck and Renner's (CR's) statement that "any hidden variable model can only be compatible with quantum mechanics if its local part is trivial" [Phys. Rev. Lett. 101, 050403 (2008)]. We note that CR's attempt to divide a nonlocal hidden variable model into a "local part" and a "nonlocal part" contains a restriction on the latter. This restriction implies that the division is really into a "local part" and a "nonsignaling nonlocal part." CR's nonsignaling requirement on the "nonlocal part" cannot be physically motivated, since the hidden variables cannot be accessed by experimenters. Nor is it a natural mathematical generalization from the local hidden variable case, since it is simple to make a generalization without CR's requirement. We give an explicit nonlocal hidden variable model that, in the case the restriction is not enforced, contains nontrivial local hidden variables.
Problem: Superposition and entanglement are coherent effects, which can be quantified by quantum mechanics (QM), but lack descriptive explanations. They are typically analysed with inequality methods, and the results favour QM and reject physical realism and hidden-variable solutions. In particular, Colbeck & Renner (2011) showed that no extension of quantum theory can exist with better predictive power than quantum mechanics itself. Purpose: The purpose here is to critically evaluate from a conceptual and philosophical perspective the ontological underpinnings of the inequality approach. The current work is speculative in nature as it is based on a conjectured non-local hidden-variable (NLHV) design for particles, and does not yet have a mathematical formalism. Nonetheless this is worth attempting for the philosophical questions it poses about the nature of reality, and the pointers it gives to possible future directions in fundamental physics. Findings: The premises of the C & R proof (that particles are points, that locality exists, that quantum theory is correct) are inconsistent, hence invalidate its conclusion. We also show that superposition and entanglement may be qualitatively explained if particles were to have the internal structure proposed by the Cordus NLHV theory. Originality: The ability to explain superposition and entanglement conceptually in terms of physical realism is relevant because it rebuts the claim that it is impossible that such a hidden-variable theory could exist. This is significant because previously it has been believed that these phenomena are explainable by QM only.
In this thesis a recent claim by Roger Colbeck and Renato Renner is reviewed [18]. This claim concerns the completeness of quantum mechanics. Colbeck and Renner state that no theory can be constructed that is compatible with quantum mechanics, and gives more detailed predictions through supplemental variables added to the description of the state of a system. Their only assumption is that measurement settings can be choosen freely. We will argue that their result is for the most part valid and interesting, but the freedom assumption is not so innocent as it is presented, and parts of the result need to be clarified. Therefore, we will present a reformulation of the result. In our formulation the result is a no-go theorem for hidden variable theories satisfying Parameter Independence. This is a generalization of Bell’s theorem. Performed in the period 01-11-2012 14-08-2012
Whether the quantum mechanics (QM) is non-local is an issue disputed for a long time. The violation of the Bell-type inequalities was considered as proving this non-locality. However, these inequalities are constructed on a class of local hidden variables, which obey the calculus with positive probabilities. Such a calculus is rather suitable for billiard balls while the QM deals with wave-packets of complex amplitudes. There is no wonder that a calculus with positive numbers does not match a calculus with complex numbers. The present text describes a different model of hidden variables for entanglements, model that reproduces the quantum predictions in different experiments, and also explains why the QM is nonlocal. The model deals with waves, some of them full and the others empty, and the hidden variables mark which waves are full. The basic physical concept with which the model operates is joint amplitudes of probability, and not probabilities. The latter are a secondary concept, the probability of a combination of results being equal to the absolute square of sum of all the contributing joint amplitudes. Thus the non-locality appears: a) a joint amplitude ignores distance, it handles distant particles as if they were one single particle at one single place, b) joint amplitudes are complex numbers and the sum of several contributions may vanish, blocking the respective combination of wave-packets and therefore of results. Although showing the success of the model, this text does not advocate for full/empty waves. It is shown that this hypothesis works only as long as one does not consider moving observers, and does not compare their conclusions. The real purpose here is to point to a severe impasse: assuming the existence of a preferred frame contradicts the theory of relativity, while refuting the full/empty waves idea one runs into other insurmountable difficulties.
Abstract We propose a novel class of local deterministic models to explain the correlations observed in entangled singlet quantum states. In these models, hidden variables influence both the entangled pair preparation and the measurement settings, thereby relaxing the measurement independence assumption used in Bell’s theorem. Our model accurately reproduces the expected quantum results and demonstrates that freely chosen measurement settings do not necessarily preclude the possibility of hidden variables inducing correlations between the settings and outcomes. This approach reveals an intricate relationship between local determinism and quantum mechanics, showing that local hidden variable theories can be consistent with the predictions of quantum theory. This result calls into question the conventional interpretation, based on Bell’s inequalities, that local theories are fundamentally incompatible with quantum mechanics.