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AIP conference proceedings

Violation of Bell’s Inequality and non-Kolmogorovness

Andrei Yu. Khrennikov, Luigi Accardi, Guillaume Adenier, Christopher A. Fuchs, Gregg S. Jaeger, Andrei Yu. Khrennikov, Jan-Åke Larsson, Stig Stenholm

Published Jan 1, 2009• 21 citations
Publisher

Abstract

In this paper we would like to stress that, besides two commonly discussed conditions inducing violation of Bell’s inequality—nonlocality and death of realism—there is the third condition having the same consequence. This is the condition of probabilistic incompatibility (PI) of random variables—impossibility to realize them on a single probability space—“non‐Kolmogorovness.” This additional source of violation of Bell’s inequality should be taken into account. We remark that PI can be a consequence of nonlocality or impossibility to use the realistic model. However, PI is essentially more general condition. Random variables can be of the PI‐type even in absence of nonolocal effects.

DOI
10.1063/1.3109976

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In this paper we would like to stress that, besides two commonly discussed conditions inducing violation of Bell’s inequality—nonlocality and death of realism—there is the third condition having the same consequence. This is the condition of probabilistic incompatibility (PI) of random variables—impossibility to realize them on a single probability space—“non‐Kolmogorovness.” This additional source of violation of Bell’s inequality should be taken into account. We remark that PI can be a consequence of nonlocality or impossibility to use the realistic model. However, PI is essentially more general condition. Random variables can be of the PI‐type even in absence of nonolocal effects.
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.
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.
The main aim of this review is to show that the common conclusion that Bell's argument implies that any attempt to proceed beyond quantum mechanics induces a nonlocal model was not totally justified. Our analysis of Bell's argument demonstrates that violation of Bell's inequality implies neither "death of realism" nor nonlocality. This violation is just a sign of non-Kolmogorovness of statistical data - impossibility to put statistical data collected in a few different experiments (corresponding to incompatible settings of polarization beam splitters) in one probability space. This inequality was well known in theoretical probability since 19th century (from works of Boole). We couple non-Kolmogorovness of data with design of modern detectors of photons.
The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell’s inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and “death of reality” which are typically linked to Bell’s type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell’s type inequalities statistical data from a number of experiments performed under different experimental contexts.
The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell's inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and "death of reality" which are typically linked to Bell's type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell's type inequalities statistical data from a number of experiments performed under different experimental contexts.