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arXiv:quant-ph/0208161 · European Journal of Physics

Assumptions underlying Bell s inequalities

Angel G Valdenebro

Published Sep 1, 2002• 22 citations
arXivHTMLPDF

Abstract

There are several versions of Bell's inequalities (BI), proved in different contexts, using different sets of assumptions. The discussions of their experimental violation often disregard some required assumptions and loosely use formulations of others. The issue, to judge from recent publications, continues to cause misunderstandings. We present a very simple but general proof of BI, identifying explicitly the complete set of assumptions required.

DOI
10.1088/0143-0807/23/5/313

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There are several versions of Bell's inequalities (BI), proved in different contexts, using different sets of assumptions. The discussions of their experimental violation often disregard some required assumptions and loosely use formulations of others. The issue, to judge from recent publications, continues to cause misunderstandings. We present a very simple but general proof of BI, identifying explicitly the complete set of assumptions required.
This article seeks to clarify certain key theoretical, conceptual and philosophical issues in the foundations of microphysics which, to judge from certain recent publications, continue to cause misunderstandings. In particular, we examine the Heisenberg indeterminacy relations, underlining that they are not univocally interpretable and that, at least in the interpretation following directly from the quantum formalism, they are not the target of Einstein, Podolsky and Rosen's criticism. We try to identify the essential goal and premises of this famous argument, with the help of a simple example. Finally, we examine briefly the Bell inequalities, emphasizing that, given their generality, the net consequence of their experimental violation cannot be circumvented neither by the abandonment of determinism nor by any local realistic reinterpretation of measurement results, as attempted in an article recently published in this journal.
Several authors have recently claimed that Bell's inequalities (BI) do not apply to certain types of generalized local hidden variables (HV) models. These claims are rejected, by means of a proof of BI valid for a very broad class of local HV models (deterministic or stochastic, with or without memory, time dependent or time independent, setting dependent or setting independent, and even allowing classical communication between the wings of the experiment). The precise meaning of the locality requirement is clarified during the proof. The exact roles of the two assumptions that amount to Bell's locality (parameter independence and outcome independence) are explicitly shown, using each one of them in a different step of the proof. using each one of them in a different step of the proof.
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.
Besides well-known conditions of locality or factorisability, deriving the Bell inequalities requires assuming that the distribution of hidden variables and Alice's and Bob's measurement settings be independent of each other. We show that (analogously to violations of locality due to action at a distance) certain violations of this Measurement Independence assumption can be associated with a notion of signalling in principle, thus making them also testable in principle, and spell out the appropriate conditions. Accordingly, we show that by imposing no-signalling one can prove a version of Bell's theorem that does not require the assumption of Measurement Independence. We discuss the "Schulman model" as an example, as well as lessons for "experimental metaphysics".