Probabilistic foundations of contextuality

Contextuality is usually defined as absence of a joint distribution for a set of measurements (random variables) with known joint distributions of some of its subsets. However, if these subsets of measurements are not disjoint, contextuality is mathematically impossible even if one generally allows (as one must) for random variables not to be jointly distributed. To avoid contradictions one has to adopt the Contextuality-by-Default approach: measurements made in different contexts are always distinct and stochastically unrelated to each other. Contextuality is reformulated then in terms of the (im)possibility of imposing on all the measurements in a system a joint distribution of a particular kind: such that any measurements of one and the same property made in different contexts satisfy a specified property, . In the traditional analysis of contextuality means “are equal to each other with probability 1”. However, if the system of measurements violates the “no-disturbance principle”, due to signaling or experimental biases, then the meaning of has to be generalized, and the proposed generalization is “are equal to each other with maximal possible probability” (applied to any set of measurements of one and the same property). This approach is illustrated on arbitrary systems of binary measurements, including most of quantum systems of traditional interest in contextuality studies (irrespective of whether the “no-disturbance” principle holds in them).

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

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

[3]  Ehtibar N. Dzhafarov,et al.  Measuring Observable Quantum Contextuality , 2015, QI.

[4]  A. Cabello,et al.  Bell-Kochen-Specker theorem: A proof with 18 vectors , 1996, quant-ph/9706009.

[5]  Ehtibar N. Dzhafarov,et al.  Context-Content Systems of Random Variables: The Contextuality-by-Default Theory , 2015, 1511.03516.

[6]  D. Kaszlikowski,et al.  Entropic test of quantum contextuality. , 2012, Physical review letters.

[7]  Garg,et al.  Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? , 1985, Physical review letters.

[8]  Anton Zeilinger,et al.  Comment on "Two Fundamental Experimental Tests of Nonclassicality with Qutrits" , 2013 .

[9]  A. Zeilinger,et al.  Experimental non-classicality of an indivisible quantum system , 2011, Nature.

[10]  Jan-AAke Larsson,et al.  Contextuality in Three Types of Quantum-Mechanical Systems , 2014, 1411.2244.

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

[12]  Ehtibar N. Dzhafarov,et al.  A Qualified Kolmogorovian Account of Probabilistic Contextuality , 2013, QI.

[13]  E. Specker DIE LOGIK NICHT GLEICHZEITIG ENTSC HEIDBARER AUSSAGEN , 1960 .

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

[15]  Patrick Suppes,et al.  When are probabilistic explanations possible? , 2005, Synthese.

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

[17]  Guido Bacciagaluppi,et al.  Leggett-Garg Inequalities, Pilot Waves and Contextuality , 2014, 1409.4104.

[18]  Ru Zhang,et al.  Is there contextuality in behavioural and social systems? , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  Jan-Åke Larsson,et al.  Necessary and Sufficient Conditions for an Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems. , 2014, Physical review letters.

[20]  R. Spekkens,et al.  Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity , 2010 .

[21]  Gary Oas,et al.  Negative probabilities and counter-factual reasoning in quantum cognition , 2014, 1404.3921.

[22]  Víctor H. Cervantes,et al.  Contextuality from Quantum Physics to Psychology , 2016 .

[23]  M. A. Can,et al.  Simple test for hidden variables in spin-1 systems. , 2007, Physical review letters.

[24]  Matt Jones,et al.  On contextuality in behavioural data , 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[25]  Ehtibar N. Dzhafarov,et al.  Probabilistic Contextuality in EPR/Bohm-type Systems with Signaling Allowed , 2014, 1406.0243.

[26]  Ehtibar N. Dzhafarov,et al.  Contextuality-by-Default 2.0: Systems with Binary Random Variables , 2016, QI.

[27]  Patrick Suppes,et al.  When are Probabilistic Explanations Possible , 1981 .

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

[29]  Thomas Filk It is the Theory Which Decides What We Can Observe (Einstein) , 2016 .

[30]  Ehtibar N. Dzhafarov,et al.  The Joint Distribution Criterion and the Distance Tests for Selective Probabilistic Causality , 2010, Front. Psychology.

[31]  G. Bacciagaluppi Einstein, Bohm, and Leggett-Garg , 2015 .

[32]  Ehtibar N. Dzhafarov,et al.  Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology , 2015, QI.

[33]  H. Thorisson Coupling, stationarity, and regeneration , 2000 .

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

[35]  J. Bell On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[36]  Ehtibar N. Dzhafarov,et al.  Proof of a Conjecture on Contextuality in Cyclic Systems with Binary Variables , 2015, 1503.02181.

[37]  Ehtibar N. Dzhafarov,et al.  Contextuality is about identity of random variables , 2014, 1405.2116.

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

[39]  Dagomir Kaszlikowski,et al.  Fundamental monogamy relation between contextuality and nonlocality. , 2013, Physical review letters.

[40]  A. Khrennikov,et al.  Epr-bohm experiment and Bell’s inequality: Quantum physics meets probability theory , 2008 .

[41]  Akihito Soeda,et al.  Generalized monogamy of contextual inequalities from the no-disturbance principle. , 2012, Physical review letters.

[42]  L. Ballentine,et al.  Quantum Theory: Concepts and Methods , 1994 .

[43]  Ehtibar N. Dzhafarov,et al.  Conversations on Contextuality , 2015 .