Contextual Probabilistic Analysis of Bell's Inequality: Nonlocality, "Death of Reality'' or Non-Kolmogorovness?

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.

[1]  Andrei Khrennikov,et al.  A perturbation of CHSH inequality induced by fluctuations of ensemble distributions , 2000 .

[2]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[3]  Itamar Pitowsky,et al.  From George Boole To John Bell — The Origins of Bell’s Inequality , 1989 .

[4]  E. Wigner.  On Hidden Variables and Quantum Mechanical Probabilities , 1970 .

[5]  Luigi Accardi,et al.  The Probabilistic Roots of the Quantum Mechanical Paradoxes , 1984 .

[6]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[7]  B. D'espagnat.  Veiled Reality: An Analysis Of Present-day Quantum Mechanical Concepts , 1995 .

[8]  Henry P. Stapp,et al.  S-MATRIX INTERPRETATION OF QUANTUM THEORY. , 1971 .

[9]  Jean-Pierre Vigier,et al.  A review of extended probabilities , 1986 .

[10]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[11]  Andrei Khrennikov,et al.  Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models , 2011 .

[12]  W. De Baere.  On the significance of Bell's inequality for hidden-variable theories , 1984 .

[13]  W. M. de Muynck,et al.  Interpretations of quantum mechanics, joint measurement of incompatible observables, and counterfactual definiteness , 1994 .

[14]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[15]  Philippe H. Eberhard.  Constraints of determinism and of Bell's inequalities are not equivalent , 1982 .

[16]  P. H. Eberhard,et al.  Bell’s theorem and the different concepts of locality , 1978 .

[17]  A. Peres.  Unperformed experiments have no results , 1978 .

[18]  Andrew Khrennikov,et al.  p-adic probability distributions of hidden variables , 1995 .

[19]  Anthony J Leggett,et al.  Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem , 2006 .

[20]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[21]  P. Lugol.  Annalen der Physik , 1906 .

[22]  F. Selleri.  Wave-Particle Duality , 2012 .

[23]  I. Pitowsky.  Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes , 1982 .

[24]  A. Zeilinger,et al.  Speakable and Unspeakable in Quantum Mechanics , 1989 .

[25]  A. Yu. Khrennikov,et al.  p‐adic quantum mechanics with p‐adic valued functions , 1991 .

[26]  Andrew Khrennikov,et al.  Statistical interpretation of p‐adic quantum theories with p‐adic valued wave functions , 1995 .

[27]  Andrei Khrennikov.  Nonlinear Schrödinger equations from prequantum classical statistical field theory , 2006 .

[28]  Jan-Åke Larsson,et al.  Quantum Paradoxes, Probability Theory, and Change of Ensemble , 2000 .

[29]  P. Pearle.  Hidden-Variable Example Based upon Data Rejection , 1970 .

[30]  Andrei Khrennikov,et al.  A pre-quantum classical statistical model with infinite-dimensional phase space , 2005, quant-ph/0505228.

[31]  Peter Rastall,et al.  The bell inequalities , 1983 .

[32]  A. Kolmogoroff.  Grundbegriffe der Wahrscheinlichkeitsrechnung , 1933 .

[33]  H. Weinfurter,et al.  Violation of Bell's Inequality under Strict Einstein Locality Conditions , 1998, quant-ph/9810080.

[34]  G. Roger,et al.  Experimental Test of Bell's Inequalities Using Time- Varying Analyzers , 1982 .

[35]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[36]  Itamar Pitowsky,et al.  Deterministic model of spin and statistics , 1983 .

[37]  Karl Hess,et al.  Bell’s theorem: Critique of proofs with and without inequalities , 2005 .

[38]  Andrei Khrennikov,et al.  Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory , 2005 .

[39]  W. M. de Muynck,et al.  On the Significance of the Bell Inequalities for the Locality Problem in Different Realistic Interpretations of Quantum Mechanics , 1988 .

[40]  T. Paterek,et al.  An experimental test of non-local realism , 2007, Nature.

[41]  Abner Shimony,et al.  Search For A Naturalistic World View , 1993 .

[42]  Andrew Khrennikov,et al.  p-adic probability interpretation of Bell's inequality , 1995 .

[43]  I. Pitowsky.  Range Theorems for Quantum Probability and Entanglement , 2001 .

[44]  Guillaume Adenier,et al.  Is the fair sampling assumption supported by EPR experiments , 2007 .

[45]  A. Shimony,et al.  Bell's theorem. Experimental tests and implications , 1978 .

[46]  Andrei Khrennikov.  Non-Kolmogorov probability models and modified Bell's inequality , 2000 .

[47]  Andrei Khrennikov,et al.  Anomalies in EPR‐Bell Experiments , 2006 .

[48]  A. Fine.  Hidden Variables, Joint Probability, and the Bell Inequalities , 1982 .

[49]  Andrei Khrennikov,et al.  Interpretations of Probability , 1999 .

[50]  Andrew Khrennikov,et al.  p-Adic stochastics and Dirac quantization with negative probabilities , 1995 .

[51]  Gregor Weihs,et al.  A Test of Bell’s Inequality with Spacelike Separation , 2007 .

[52]  C. Ross.  Found , 1869, The Dental register.

[53]  N. N. Vorob’ev.  Consistent Families of Measures and Their Extensions , 1962 .

[54]  Luigi Accardi,et al.  Could we now convince Einstein , 2006 .