CHSH Inequality: Quantum Probabilities as Classical Conditional Probabilities

In this note we demonstrate that the results of observations in the EPR–Bohm–Bell experiment can be described within the classical probabilistic framework. However, the “quantum probabilities” have to be interpreted as conditional probabilities, where conditioning is with respect to fixed experimental settings. Our approach is based on the complete account of randomness involved in the experiment. The crucial point is that randomness of selections of experimental settings has to be taken into account within one consistent framework covering all events related to the experiment. This approach can be applied to any complex experiment in which statistical data are collected for various (in general incompatible) experimental settings.

[1]  Andrei Khrennikov,et al.  On the equivalence of the Clauser–Horne and Eberhard inequality based tests , 2014, 1403.2811.

[2]  L. Accardi.  Topics in quantum probability , 1981 .

[3]  Andrei Khrennikov,et al.  Ubiquitous Quantum Structure , 2010 .

[4]  Caslav Brukner,et al.  Information Invariance and Quantum Probabilities , 2009, 0905.0653.

[5]  The 'symplectic camel principle' and semiclassical mechanics , 2002, math/0207216.

[6]  Caslav Brukner,et al.  Experimenter's freedom in Bell's theorem and quantum cryptography (7 pages) , 2005, quant-ph/0510167.

[7]  Marian Kupczynski,et al.  EPR Paradox, Locality and Completeness of Quantum Theory , 2007, 0710.3510.

[8]  Andrei Khrennikov,et al.  Single, Complete, Probability Spaces Consistent With EPR‐Bohm‐Bell Experimental Data , 2009 .

[9]  G. D’Ariano.  Philosophy of Quantum Information and Entanglement: Probabilistic theories: What is special about Quantum Mechanics? , 2008, 0807.4383.

[10]  K. Michielsen,et al.  A local realist model for correlations of the singlet state , 2006 .

[11]  Andrei Khrennikov.  Quantum theory: Reconsideration of foundations , 2003 .

[12]  Aaron D. O’Connell.  Dance of the Photons: From Einstein to Quantum Teleportation , 2011 .

[13]  Andrei Khrennikov,et al.  Classical probability model for Bell inequality , 2014, 1404.7038.

[14]  Willem M. de Muynck.  Interpretations of quantum mechanics, and interpretations of violation of Bell's inequality , 2001 .

[15]  M. Kupczynski.  Entanglement and quantum nonlocality demystified , 2012, 1205.4636.

[16]  Igor Volovich,et al.  Quantum field theory and distance effects for polarization correlations in waveguides , 2009 .

[17]  Karl Hess,et al.  Exclusion of Time in Mermin's Proof of Bell-Type Inequalities , 2003 .

[18]  Andrei Khrennikov,et al.  Kolmogorov Probability Spaces Describing Accardi Models of Quantum Correlations , 2005, Open Syst. Inf. Dyn..

[19]  Classical-like description of quantum dynamics by means of symplectic tomography , 1996, quant-ph/9609026.

[20]  Luigi Accardi,et al.  Locality and Bell's inequality , 2000, quant-ph/0007005.

[21]  A. Peres,et al.  Probabilistic Theories: What Is Special about Quantum Mechanics? , 2008 .

[22]  E. Dzhafarov.  Selective influence through conditional independence , 2003 .

[23]  A. Zeilinger,et al.  Bell violation using entangled photons without the fair-sampling assumption , 2012, Nature.

[24]  Sergey V. Polyakov,et al.  Quantum Theory: Reconsideration of Foundations-4 , 2007 .

[25]  W. D. Muynck.  Foundations of Quantum Mechanics, an Empiricist Approach , 2002 .

[26]  Andrei Khrennikov,et al.  Foundations of Probability and Physics , 2002 .

[27]  Ehtibar N. Dzhafarov,et al.  Embedding Quantum into Classical: Contextualization vs Conditionalization , 2013, PloS one.

[28]  S. Miyashita,et al.  Event-Based Computer Simulation Model of Aspect-Type Experiments Strictly Satisfying Einstein's Locality Conditions , 2007, 0712.2565.

[29]  Guillaume Adenier,et al.  Local Realist Approach and Numerical Simulations of Nonclassical Experiments in Quantum Mechanics , 2008 .

[30]  Ehtibar N. Dzhafarov,et al.  No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics , 2014 .

[31]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

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

[33]  Luigi Accardi,et al.  Some loopholes to save quantum nonlocality , 2005 .

[34]  Caslav Brukner,et al.  OPERATIONALLY INVARIANT INFORMATION IN QUANTUM MEASUREMENTS , 1999 .

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

[36]  G. D’Ariano,et al.  Informational derivation of quantum theory , 2010, 1011.6451.

[37]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[38]  Caslav Brukner,et al.  Condition for macroscopic realism beyond the Leggett-Garg inequalities , 2012, 1207.3666.

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

[40]  M. Kupczyński,et al.  Bertrand's paradox and Bell's inequalities , 1987 .

[41]  Yoshiharu Tanaka,et al.  Violation of contextual generalization of the Leggett–Garg inequality for recognition of ambiguous figures , 2014, 1401.2897.

[42]  Marian Kupczynski.  Seventy Years of the EPR Paradox , 2006 .

[43]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

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

[45]  A. Zeilinger.  A Foundational Principle for Quantum Mechanics , 1999, Synthese Library.

[46]  R. Mcweeny.  On the Einstein-Podolsky-Rosen Paradox , 2000 .

[47]  Marian Kupczynski,et al.  Causality and local determinism versus quantum nonlocality , 2013, 1312.0636.

[48]  Andrei Khrennikov,et al.  Frequency Analysis of the EPR-Bell Argumentation , 2002 .

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

[50]  Igor Volovich,et al.  Photon Flux and Distance from the Source: Consequences for Quantum Communication , 2013, 1309.5492.

[51]  V. I. Man'ko,et al.  Classical formulation of quantum mechanics , 1996 .

[52]  Andrei Khrennikov,et al.  Ubiquitous Quantum Structure: From Psychology to Finance , 2010 .

[53]  Sven Nordebo,et al.  Distance dependence of entangled photons in waveguides , 2012 .

[54]  Aaron J. Miller,et al.  Detection-loophole-free test of quantum nonlocality, and applications. , 2013, Physical review letters.

[55]  B. Lavenda.  Classical formulations of quantum-mechanical time-dependent variational principles , 2003 .

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

[57]  Ehtibar N. Dzhafarov,et al.  On Selective Influences, Marginal Selectivity, and Bell/CHSH Inequalities , 2012, Top. Cogn. Sci..

[58]  Andrei Khrennikov,et al.  Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables? , 2008, Entropy.

[59]  Andrei Khrennikov,et al.  Contextual Approach to Quantum Formalism , 2009 .