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arXiv:quant-ph/0004037 · Il Nuovo Cimento B

"Violating'' Clauser-Horne inequalities within classical mechanics

S. Antoci, Luigi Mihich

Published Apr 7, 2000• 0 citations
arXivHTMLPDF

Abstract

Some authors have raised the question whether the probabilities stemming from a quantum mechanical computation are entitled to enter the Bell and the Clauser-Horne inequalities. They have remarked that if the quantum probabilities are given the status of conditional ones and the statistics for the various settings of the detectors in a given experiment is properly kept into account, the inequalities happen to be no longer violated. In the present paper a classical simile modeled after the quantum mechanical instances is closely scrutinised. It is shown that the neglect of the conditional character of the probabilities in the classical model leads not only to ``violate'' the Clauser-Horne inequalities, but also to contradict the very axioms of classical probability theory.

DOI
10.48550/arxiv.quant-ph/0004037

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Some authors have raised the question whether the probabilities stemming from a quantum mechanical computation are entitled to enter the Bell and the Clauser-Horne inequalities. They have remarked that if the quantum probabilities are given the status of conditional ones and the statistics for the various settings of the detectors in a given experiment is properly kept into account, the inequalities happen to be no longer violated. In the present paper a classical simile modeled after the quantum mechanical instances is closely scrutinised. It is shown that the neglect of the conditional character of the probabilities in the classical model leads not only to ``violate'' the Clauser-Horne inequalities, but also to contradict the very axioms of classical probability theory.
This paper intends to show how the fabled violation of Bell's inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect.I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt.Designed to assess consequences of Einstein's principle of local realism, the inequality pertains to a li-How to cite this paper: Lad, F. ( 2021) Quantum Violation of Bell's Inequality: A Misunderstanding Based on a Mathematical Error of Neglect.
We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative -- either nonlocality or ``death of reality'' (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables -- impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality -- a few pairs of random variables (two dimensional vector variables) involved in the EPR-Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus one can choose between: a) completeness of quantum mechanics; b) nonlocality; c) `` death of reality''; d) non-Kolmogorovness. In any event, violation of Bell's inequality has a variety of possible interpretations. Hence, it could not be used to derive the unique conclusion on the relation between quantum and classical models.
Most physicists uphold that the tests of the Bell-Clauser, Horne, Shimony and Holt (BCHSH) inequalities confirm quantum mechanics (QM) and refute local realism. Some scholars, however, criticize this conviction, yet maintaining that QM conflicts with local realism. At variance with both viewpoints one of the authors has recently worked out an extended semantic realism (ESR) model which shows that, if quantum probabilities are interpreted as conditional rather than absolute probabilities, a picture of the microworld can be constructed that is simultaneously consistent with local realism and QM. We show here that the BCHSH inequalities must be replaced, within the ESR model, by more general inequalities. These depend on parameters (detection probabilities) which may be such that the inequalities are never violated by quantum expectation values. The condition that no violation occurs implies the existence of upper bounds on detection probabilities, which makes the ESR model falsifiable. These results admit an intuitive explanation in terms of unfair sampling but basically differ from the seemingly similar results obtained by other approaches in which the efficiency problem is discussed in order to vindicate some kind of local realism.
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.
It is argued that any quantum probability should correspond to a ratio between the number of counts (positive results) in some measurement and the number of copies of the physical system initially prepared. Then, the proofs of Bell's theorem are criticized on the grounds that the probabilities used to show a violation of the Bell inequalities do not fulfill that condition. A hidden-variables model is proposed which reproduces the results of the optical experimental tests of the inequalities, even with perfect polarizers and detectors.