An Appropriate Probability Model for the Bell Experiment

TL;DR

The paper argues that the Bell-CHSH contradiction arises from using an implicit four-observable probability model and shows that an explicit two-observable framework with conditional quantum expectations $\mathbb{E}[XY|A=a,B=b]$ aligns with quantum mechanics. By constructing a probability space with two observables and two detector settings, the authors recover the Born-rule predictions $p(x,y|\alpha,\beta)$ and $\mathbb E[XY|A,B]$, avoiding the unconditional CHSH bound violation. They demonstrate statistical locality ($\mathbb P[X=x|A,B]=\tfrac12$) while proving non-separability of the joint measure, ruling out local hidden-variable theories and implying that locality and realism cannot both hold in a separable form. Consequently, the usual interpretation of CHSH violation as evidence against local realism is unfounded within this QM-consistent probabilistic framework, and the results emphasize the role of conditional expectations and non-separability in reconciling quantum correlations with probability theory.

Abstract

The Bell inequality constrains the outcomes of measurements on pairs of distant entangled particles. The Bell contradiction states that the Bell inequality is inconsistent with the calculated outcomes of these quantum experiments. This contradiction led many to question the underlying assumptions, viz. so-called realism and locality. The probability model underlying the Bell inequality is generally left implicit. This implicit consensus model assumes four simultaneously observable detector settings. The Bell contradiction follows from this assumption. We propose an explicit probability model for the CHSH version of the Bell experiment. This model has only two simultaneously observable detector settings per measurement, and therefore does not assume realism. The quantum expectation now becomes a conditional expectation, given the two detector settings. This probability model is in full agreement with both quantum mechanics and experiments. In this model the notion of Bell contradiction has no meaning. Furthermore, the proposed probability model is statistically local, and is not Bell-separable. The latter implies that either hidden variables must be ruled out, or that locality must be violated, in agreement with Bell's conclusion.

Figures (1)

• Figure 1: The optical version of the Bell experiment. The entangled photon pair is described by wave function $\Psi$. The polarimeters $X_\alpha$ and $Y_\beta$ each comprise a polarizer, with rotatable polarization angles $\alpha$ and $\beta$, and two photon counters labeled D+ and D-. The coincidence monitor $\boldsymbol{{\langle}} \Psi\; | \; X_\alpha \otimes Y_\beta \; | \; \Psi \boldsymbol{{\rangle}}$ computes the corresponding quantum mechanical expectation.