Statistics, Causality and Bell's Theorem

Abstract: Bell's [Physics 1 (1964) 195-200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell's inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804-1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell's inequality and thereby also of Bell's theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom. Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the actual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Methodological and statistical issues in the design of quantum Randi challenges (QRC) are discussed. The paper argues that Bell's theorem (and its experimental confirmation) should lead us to relinquish not locality, but realism.

Statistics, Causality and Bell's Theorem: A Review

The paper "Statistics, Causality, and Bell's Theorem" by Richard D. Gill examines the implications of Bell's theorem in the context of quantum mechanics, particularly focusing on its interplay with statistical causality concepts. It enriches the discussion by providing a strong finite sample version of Bell's inequality and elucidating the incompatibility of quantum mechanics with classical principles of locality, realism, and freedom.

Bell's theorem, introduced by John Bell in 1964, challenges the classical notions by demonstrating that the predictions of quantum mechanics cannot be reconciled with the conjunction of locality, realism, and freedom. Locality asserts that causal influences need time to propagate, realism concerns itself with counterfactual reasoning, and freedom relates to the randomness in experimental choices. The violation of Bell's inequality, experimentally evidenced by Aspect et al. and Weihs et al., manifests this fundamental incompatibility, suggesting that at least one classical principle must be abandoned.

Gill's version of Bell's inequality, presented in the paper, is distinct in that it operates directly on finite sample averages rather than theoretical expectations. This approach provides a practical framework for evaluating the results of empirical tests of Bell-type experiments. The paper details how traditional Bell-CHSH inequalities fail to hold in quantum mechanics' contexts due to the inherent quantum correlations that result in violations expected under certain configurations, notably exceeding classical boundaries, like achieving values up to 2√2, contrary to the classical limit of 2.

The experiment described uses entangled particles—often photons—measured at distant locations in varied settings to demonstrate the violation of Bell inequalities. This empirical foundation complements the theoretical underpinnings, deeply rooted in quantum mechanics' principles. Despite advancements in experimental techniques, challenges such as detection and locality loopholes persist, necessitating meticulous experimental designs to ensure fidelity to these quantum predictions.

Gill argues for rejecting realism rather than locality, positing that quantum randomness should be seen as a fundamental aspect of reality, not merely an emergent property. This proposition entails accepting that measurement outcomes are not predetermined and that the act of measurement itself generates these outcomes. Realism, as incorporated in classical physics, assumes the existence of outcomes even for unperformed measurements—a view quantum mechanics directly challenges.

Future developments in closing experimental loopholes are critical for solidifying the empirical confirmation of Bell's theorem. The paper accentuates the importance of understanding the philosophical implications behind rejecting realism, which requires adjusting our perception of causality, given our cognitive predispositions toward deterministic interpretations of physical phenomena.

In conclusion, the synthesis of statistical causality and quantum mechanics in investigating Bell's theorem enriches both theoretical and practical understandings of quantum phenomena. The paper anticipates continued progress in empirical tests and the refinement of theoretical models that could someday lead to a more comprehensive reconciliation of classical and quantum worldviews.