Indistinguishability and Negative Probabilities

In this paper, we examined the connection between quantum systems’ indistinguishability and signed (or negative) probabilities. We do so by first introducing a measure-theoretic definition of signed probabilities inspired by research in quantum contextuality. We then argue that ontological indistinguishability leads to the no-signaling condition and negative probabilities.

[1]  J. Acacio de Barros Beyond the quantum formalism: consequences of a neural-oscillator model to quantum cognition , 2013, ArXiv.

[2]  K. E. CAHnL Density Operators and Quasiprobability Distributions * , 2011 .

[3]  Jose Acacio de Barros,et al.  Decision Making for Inconsistent Expert Judgments Using Negative Probabilities , 2013, QI.

[4]  Giuseppe Compagno,et al.  Indistinguishability of Elementary Systems as a Resource for Quantum Information Processing. , 2017, Physical review letters.

[5]  Patrick Suppes,et al.  Existence of hidden variables having only upper probabilities , 1991 .

[6]  G. Compagno,et al.  Dealing with indistinguishable particles and their entanglement , 2018, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[9]  Newton C. A. da Costa,et al.  A modal ontology of properties for quantum mechanics , 2012, Synthese.

[10]  F. Holik,et al.  A Discussion on Particle Number and Quantum Indistinguishability , 2007, 0705.3417.

[11]  Federico Holik,et al.  Contextuality and Indistinguishability , 2017, Entropy.

[12]  Armin W. Schulz,et al.  Interpretations of probability , 2003 .

[13]  Maria Carla Galavotti Philosophical introduction to probability , 2005 .

[14]  Robert Stalnaker Context and content : essays on intentionality in speech and thought , 1999 .

[15]  Augustus De Morgan On the study and difficulties of mathematics , 1898 .

[16]  Mark Burgin,et al.  Interpretations of Negative Probabilities , 2010, 1008.1287.

[17]  Ehtibar N. Dzhafarov,et al.  Snow Queen Is Evil and Beautiful: Experimental Evidence for Probabilistic Contextuality in Human Choices , 2017, Decision.

[18]  R. Feynman The development of the space-time view of quantum electrodynamics. , 1966, Science.

[19]  J. A. Tenreiro Machado,et al.  Fractional coins and fractional derivatives , 2013 .

[20]  Patrick Suppes,et al.  Probabilistic inequalities and upper probabilities in quantum mechanical entanglement , 2010, 1010.3064.

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

[22]  Robert W Spekkens,et al.  Negativity and contextuality are equivalent notions of nonclassicality. , 2006, Physical review letters.

[23]  Werner Stulpe,et al.  Phase‐space representations of general statistical physical theories , 1992 .

[24]  Ehtibar N. Dzhafarov,et al.  Unifying Two Methods of Measuring Quantum Contextuality , 2014 .

[25]  J. A. Barros,et al.  Quantum Cognition, Neural Oscillators, and Negative Probabilities , 2017 .

[26]  Samson Abramsky,et al.  An Operational Interpretation of Negative Probabilities and No-Signalling Models , 2014, Horizons of the Mind.

[27]  Daniel Gottesman,et al.  Classicality in discrete Wigner functions , 2005, quant-ph/0506222.

[28]  Gary Oas,et al.  Negative Probabilities and Contextuality , 2015, 1511.02823.

[29]  Décio Krause,et al.  On a Quasi-Set Theory , 1992, Notre Dame J. Formal Log..

[30]  E. Galvão Discrete Wigner functions and quantum computational speedup , 2004, quant-ph/0405070.

[31]  David J. Foulis,et al.  Properties and operational propositions in quantum mechanics , 1983 .

[32]  Ehtibar N. Dzhafarov,et al.  The Contextuality-by-Default View of the Sheaf-Theoretic Approach to Contextuality , 2019, 1906.02718.

[33]  J. Acacio de Barros,et al.  Exploring non-signalling polytopes with negative probability , 2014, 1404.3831.

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

[35]  Erwin Schrödinger Science And Humanism , 1951 .

[36]  W. Zurek Environment-induced superselection rules , 1982 .

[37]  Arthur Fine,et al.  Joint distributions, quantum correlations, and commuting observables , 1982 .

[38]  P. Dirac,et al.  The physical interpretation of quantum mechanics , 1942 .

[39]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[40]  M. Scully,et al.  Distribution functions in physics: Fundamentals , 1984 .

[41]  The Bakerian Lecture , 2022, Nature.

[42]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[43]  Gary Oas,et al.  A Survey of Physical Principles Attempting to Define Quantum Mechanics , 2015, 1506.05515.

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

[45]  C. H. Randall,et al.  Operational Statistics. I. Basic Concepts , 1972 .

[46]  Federico Holik,et al.  A discussion on the origin of quantum probabilities , 2012, 1211.4952.

[47]  F. Holik,et al.  Indistinguishability and the origins of contextuality in physics , 2019, Philosophical Transactions of the Royal Society A.

[48]  Samson Abramsky,et al.  The sheaf-theoretic structure of non-locality and contextuality , 2011, 1102.0264.

[49]  Patrick Suppes,et al.  Probability Concepts in Quantum Mechanics , 1961, Philosophy of Science.

[50]  F. Holik,et al.  Q-spaces and the Foundations of Quantum Mechanics , 2008, 0803.4517.

[51]  M. Horne,et al.  Experimental Consequences of Objective Local Theories , 1974 .

[52]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[53]  Anthony J Short,et al.  Simulating all nonsignaling correlations via classical or quantum theory with negative probabilities. , 2013, Physical review letters.

[54]  A. Cabello Simple explanation of the quantum violation of a fundamental inequality. , 2012, Physical review letters.

[55]  Ehtibar N. Dzhafarov,et al.  Generalizing Bell-type and Leggett-Garg-type Inequalities to Systems with Signaling , 2014, 1407.2886.

[56]  Leonhardt,et al.  Discrete Wigner function and quantum-state tomography. , 1996, Physical review. A, Atomic, molecular, and optical physics.

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

[58]  S. Popescu,et al.  Quantum nonlocality as an axiom , 1994 .

[59]  Samson Abramsky,et al.  Logical Bell Inequalities , 2012, ArXiv.

[60]  J. Emerson,et al.  Corrigendum: Negative quasi-probability as a resource for quantum computation , 2012, 1201.1256.

[61]  Acacio De Barros J,et al.  Inequalities for dealing with detector inefficiencies in greenberger-horne-zeilinger-type experiments , 2000, Physical review letters.

[62]  Yutaka Shikano,et al.  Strange weak values , 2010, 1006.1615.

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

[64]  D. Dieks Communication by EPR devices , 1982 .

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

[66]  Kapuściński Wj,et al.  Science and Humanism , 1935, Nature.

[67]  Paul Adrien Maurice Dirac,et al.  Bakerian Lecture - The physical interpretation of quantum mechanics , 1942, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

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

[69]  Mark Burgin,et al.  An Introduction to Symmetric Inflated Probabilities , 2016, QI.

[70]  D. Krause,et al.  Quasi-set theory for bosons and fermions: Quantum distributions , 1999 .

[71]  Andreas Blass,et al.  Negative probability , 1945, Mathematical Proceedings of the Cambridge Philosophical Society.

[72]  F. Holik,et al.  No Labeling Quantum Mechanics of Indiscernible Particles , 2009, 0904.3476.