Quantum Violation of Bell’s Inequality: A Misunderstanding Based on a Mathematical Error of Neglect

Abstract

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 linear combination of four polarization products on the same pair of photons arising in a gedankenexperiment. My assessment displays that in this context, the summands of the relevant CHSH quantity s ( λ ) inhere four symmetric functional relations which have long been neglected in analytic considerations. Its expectation E[ s ( λ ) ] is not the sum of four “marginal” expectations from a joint distribution, as quantum theory explicitly avoids such a specification. Rather, I show that E[ s ( λ ) ] has four distinct representations as the sum of three expectations of polarization products plus the expectation of a fourth which is restricted to equal a function value determined by the other three. Analysis using Bruno de Finetti’s fundamental theorem of prevision (FTP) yields only a bound for E ( s ) within (1.1213,2] , surely not at all as is commonly understood. I exhibit slices of the 4-dimensional polytope of joint P ++ probabilities actually motivated by quantum theory at the four stipulated angle settings, as it passes through 3-dimensional space. Bell’s inequality is satisfied everywhere within the convex hull of extreme distributions cohering with quantum theoretic specifications, even while in keeping with local realism. Aspect’s proposed “estimation” of E ( s ) near to is based on polarization products from different photon pairs that do not have embedded within them the functional relations inhering in the relevant gedankenexperiment. When one actively embeds the restrictions into Aspect’s estimation procedure, it yields an estimate of 1.7667, although this is not and cannot be definitive. While my analysis supports the subjectivist construction of probability as clarifying issues relevant to the interpretation of quantum theory, the error resolved herein is purely mathematical. It pertains to the reconsideration of Bell violation irrespective of one’s attitude toward the meaning of probability.