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Proceedings of the National Academy of Sciences

Breakdown of Bell's theorem for certain objective local parameter spaces

K. Hess, Walter Philipp

Published Jan 22, 2004• 40 citations
Publisher

Abstract

We show that the known proofs of Bell's inequalities contain algebraic manipulations that are not appropriate within the syntax of Kolmogorov's axioms for probability theory without detailed justification. Such justification can be achieved by a variant of the techniques used in Bell-type proofs but only for a subclass of objective local parameter spaces. It cannot be achieved for an extended parameter space that is still objective local and that includes instrument parameters correlated by both time and setting dependencies.

DOI
10.1073/pnas.0307479100

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We show that the known proofs of Bell's inequalities contain algebraic manipulations that are not appropriate within the syntax of Kolmogorov's axioms for probability theory without detailed justification. Such justification can be achieved by a variant of the techniques used in Bell-type proofs but only for a subclass of objective local parameter spaces. It cannot be achieved for an extended parameter space that is still objective local and that includes instrument parameters correlated by both time and setting dependencies.
Most of the standard proofs of the Bell theorem are based on the Kolmogorov axioms of probability theory. We show that these proofs contain mathematical steps that cannot be reconciled with the Kolmogorov axioms. Specifically we demonstrate that these proofs ignore the conclusion of a theorem of Vorob'ev on the consistency of joint distributions. As a consequence Bell's theorem stated in its full generality remains unproven, in particular, for extended parameter spaces that are still objective local and that include instrument parameters that are correlated by both time and instrument settings. Although the Bell theorem correctly rules out certain small classes of hidden variables, for these extended parameter spaces the standard proofs come to a halt. The Greenberger-Horne-Zeilinger (GHZ) approach is based on similar fallacious arguments. For this case we are able to present an objective local computer experiment that simulates the experimental test of GHZ performed by Pan, Bouwmeester, Daniell, Weinfurter and Zeilinger and that directly contradicts their claim that Einstein-local elements of reality can neither explain the results of quantum mechanical theory nor their experimental results.
The celebrated inequalities of Bell are based on the assumption that local hidden parameters exist. When combined with conflicting experimental results, these inequalities appear to prove that local hidden parameters cannot exist. This contradiction suggests to many that only instantaneous action at a distance can explain the Einstein, Podolsky, and Rosen type of experiments. We show that, in addition to the assumption that hidden parameters exist, Bell tacitly makes a variety of other assumptions that contribute to his being able to obtain the desired contradiction. For instance, Bell assumes that the hidden parameters do not depend on time and are governed by a single probability measure independent of the analyzer settings. We argue that the exclusion of time has neither a physical nor a mathematical basis but is based on Bell's translation of the concept of Einstein locality into the language of probability theory. Our additional set of local hidden variables includes time-like correlated parameters and a generalized probability density. We prove that our extended space of local hidden variables does not permit Bell-type proofs to go forward.
By implicitly assuming that all possible Bell-measurements occur simultaneously, all proofs of Bell's Theorem violate Heisenberg's Uncertainty Principle. This assumption is made in the original form of Bell's inequality, in Wigner's probability inequalities, and in the ``nonlocality without inequalities'' arguments. The introduction of time into derivations of these variants of Bell's theorem results in extra terms related to the time order of the measurements used in constructing correlation coefficients. Since the same locality assumptions are made in the Heisenberg-compliant derivations of this paper, only time-independent classical local hidden variable theories are forbidden by violations of the original Bell inequalities; time-dependent quantum local hidden variable theories can satisfy this new bound and agree with experiment. We further point out that factorizable wavefunctions have been used to describe some EPR experiments and can be used to describe others. These will generate local de Broglie-Bohm trajectories in the description of the data. This second, independent, line of argument also shows that violation of Bell's inequality is only evidence that Heisenberg's Uncertainty Principle cannot be ignored.
Among the various "no-go theorems" in the foundations of quantum mechanics, Bell inequalities are among the few, which do not implicitly rely on the mathematical structure of quantum mechanics. Instead, Bell inequalities are commonly understood to rest on Kolmogorov's theory of probability and a "locality" assumption. Physical evidence supporting their violation is hence taken as evidence against at least one of these two assumptions. This work provides a comprehensive analysis of the physical debate on Bell inequalities from the point of view of Kolmogorov's theory of probability, focusing on the Bell-Clauser-Horn-Shimony-Holt inequality (BCHSH inequality). The central contribution is the motivation and mathematical study of the notion of contextuality in Bell experiments: in so-called contextual Bell models the detectors themselves behave randomly and the distribution of the random detector parameters depends on the chosen detector settings. It is shown how even "classical" Bell experiments motivate such a description. The need to account for contextuality was pointed out before by de la Peña, Cetto, Brody, Lochak, Bohm, Hiley and others. A broad class of contextual models is proven to satisfy the BCHSH inequality, thereby generalizing a previous result by Gill and Lambare. Nonetheless, even under strong locality requirements, contextual Bell models may violate the BCHSH inequality: based on work by Popescu and Rohrlich, we consider a "proof of concept" example exhibiting the maximum possible violation. Contrary to the much debated "loopholes" in empirical tests of Bell inequalities, contextuality thus reveals a conceptual misunderstanding in the physical conclusions commonly drawn from Bell inequality violations.