On some class of random variables leading to violations of the Bell inequality

Enrico G. Beltrametti, Macie J. Ma̧czyński

V. A. Andreev

The Bell inequality two-particle spin states are considered. It is shown that violation of this inequality at experimental verifications is connected with the fact that it is proved for some arbitrary random variables, but in experimental verification random variables of special type are used. A new inequality is constructed. It contains a correlation coefficient of random variables, measured at the experiment, and does not have to be violated at experimental verification. For factorizable and separable states it coincides with the usual Bell inequality.

Andrei Yu. Khrennikov, Guillaume Adenier, Andrei Yu. Khrennikov +3 more

We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative—either nonlocality or “death of reality” (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables—impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality—a few pairs of random variables (two dimensional vector variables) involved in the EPR‐Bohm experiment are incompatible. They could not be realized on a single Kolmogorov probability space. Thus one can choose between: a) completeness of quantum mechanics; b) nonlocality; c) “death of reality”; d) non‐Kolmogorovness. In any event, violation of Bell's inequality has a variety of possible interpretations. Hence, it could not be used to derive the unique conclusion on the relation between quantum and classical models.

Loran V. Akopyan, Vladimir I. Man’ko

Adam Bednorz, Jakub Zieliński

Patrick Suppes, Mario Zanotti

INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.