Bell's theorem and Bayes' theorem

Arthur J. Parzygnat, Benjamin P. Russo

Sheldon Goldstein, Travis Norsen, Daniel V. Tausk +1 more

Robert A. Close

Bell's Theorem places limits on correlations between local spin measurements of entangled particles whose properties are described by hidden variables established prior to measurement. Bell's derivation assumes that the density of states, or sampling rate, is independent of the orientation of the measuring device. However, points on a rotating sphere are sampled at different rates at different positions, making Bell's Theorem inapplicable. We model spin one-half fermions as having a spherical distribution of observables with azimuthal symmetry, and assume that a Stern-Gerlach device uniformly samples points on a spherical surface. Application of Bayes' Theorem yields the joint density of states for two device orientations. Numerical calculations based on this model yield the fermion spin correlations observed in Stern-Gerlach experiments. c

Richard David Gill

In this short note, I derive the Bell-CHSH inequalities as an elementary result in the present-day theory of statistical causality based on graphical models or Bayes' nets, defined in terms of DAGs (Directed Acyclic Graphs) representing direct statistical causal influences between a number of observed and unobserved random variables. I show how spatio-temporal constraints in loophole-free Bell experiments, and natural classical statistical causality considerations, lead to Bell's notion of local hidden variables, and thence to the CHSH inequalities. The word “local” applies to the way that the chosen settings influence the observed outcomes. The case of contextual setting-dependent hidden variables (thought of as being located in the measurement devices and dependent on the measurement settings) is automatically covered, despite recent claims that Bell's conclusions can be circumvented in this way.

Anthony J. Leggett

Matthias Schultheiss