Comparison between local Kolmogorovian models and Bell's inequality

David Atkinson

Adam Stokes

Within the Dempster-Schafer theory of evidence a non-Kolmogorovian kind of epistemic uncertainty arises, which is encoded using multi-valued maps. We analyse the possible implications such non-Kolmogorovian epistemic uncertainty may have for Bell-type inequalities relating to the Einstein-Podolsky-Rosen-Bohm (EPRB) thought experiment. Our analysis leads to a notion of contextuality concerning complexes of physical measurement conditions. The use of multi-valued maps reveals an implicit link between this contextuality and counterfactual outcomes, and results in a formulation wherein the states of measurement devices are explicitly taken into account as part of the probabilistic event space. This reflects a conception of measurement that was advocated by Bell some time ago. It results in context-conditioned measure-theoretic probabilities, which do not obey Bell-type inequalities, but which are nonetheless perfectly compatible with local classical physical models. We give an example of a local classical model that reproduces the quantum mechanical predictions and that fits within the contextual framework.

Jarosław Pykacz

Bell-type inequalities, used in mathematical physics as a criterion to check whether a physical situation allows description in terms of classical (Kolmogorovian) or quantum probability calculus are applied to various fuzzy probability models. It occurs that the standard set of Bell-type inequalities does not allow to distinguish Kolmogorovian probabilities from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows to distinguish all these models from fuzzy probability models based on Giles (Łukasiewicz) intersection. It is proved that if we use fuzzy set intersections pointwisely generated by Frank's fundamental triangular norms Ts(x,y), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and these fuzzy probability models that cannot be distinguished is for .

Yalçin Koç

Johannes F. Geurdes

The statistics behind Bell's inequality is demonstrated to allow a Kolmogorovian (i.e. classical) model of probabilities that recovers the quantum covariance.