Bell-type inequalities in classical probability theory

Peter Millington, Zong-Gang Mou, Paul M. Saffin +1 more

Abstract Inspired by Lefschetz thimble theory, we treat Quantum Field Theory as a statistical theory with a complex Probability Distribution Function (PDF). Such complex-valued PDFs permit the violation of Bell-type inequalities, which cannot be violated by a real-valued, non-negative PDF. In this paper, we consider the Classical-Statistical approximation in the context of Bell-type inequalities, viz. the familiar (spatial) Bell inequalities and the temporal Leggett-Garg inequalities. We show that the Classical-Statistical approximation does not violate temporal Bell-type inequalities, even though it is in some sense exact for a free theory, whereas the full quantum theory does. We explain the origin of this discrepancy, and point out the key difference between the spatial and temporal Bell-type inequalities. We comment on the import of this work for applications of the Classical-Statistical approximation.

Enrico G. Beltrametti, Macie J. Ma̧czyński

Enrico G. Beltrametti, Maciej Mączyński

Loran V. Akopyan, Vladimir I. Man’ko

Multi-qudit systems are studied in tomographic probability representations of quantum qudit states. Results of calculations for Bell-type numbers within the framework of classical probability theory and in quantum tomography are compared. Violation of Bell-type inequalities are shown explicitly using the method of averaging in tomographic picture of quantum states.

Thomas V Marcella

Andrei Yu. Khrennikov

We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be defined by classical Bayes' formula. We also use the approach based on Bell-type inequalities in the conventional probabilistic approach, Kolmogorov model. We prove an analog of Wigner's inequality for conditional probabilities and by using this inequality show that predictions of the conventional and quantum probability models disagree already in the case of noncomposite systems (even in the two dimensional case).