Varieties of contextuality based on probability and structural nonembeddability. (English) Zbl 1543.81019
Summary: Different analytic notions of contextuality fall into two major groups: probabilistic and strong notions of contextuality. S. Kochen and E. P. Specker’s Theorem 0 [J. Math. Mech. 17, 59–87 (1967; Zbl 0156.23302)] presents a demarcation criterion for differentiating between those groups. Whereas probabilistic contextuality still allows classical models, albeit with nonclassical probabilities, the logico-algebraic “strong” form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Both forms indicate a classical in- or under-determination that can be termed “value indefinite” and formalized by partial functions of theoretical computer sciences.
MSC:
| 81P13 | Contextuality in quantum theory |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 81P40 | Quantum coherence, entanglement, quantum correlations |
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
| 05C60 | Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) |
Keywords:
quantum randomness; Gleason theorem; Kochen-Specker theorem; Born rule; object construction; quantum entanglement; “value indefinite”Citations:
Zbl 0156.23302References:
| [1] | Kochen, S.; Specker, E. P., The problem of hidden variables in quantum mechanics, J. Math. Mech., 17, 1, 59-87 (1967) · Zbl 0156.23302 |
| [2] | Boole, G., On the theory of probabilities, Philos. Trans. R. Soc. Lond., 152, 225-252 (1862) · Zbl 1319.60004 |
| [3] | Schrödinger, E., Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften, 23, 823-828 (1935) · JFM 61.0936.01 |
| [4] | Bohr, N., Discussion with Einstein on epistemological problems in atomic physics, (Schilpp, P. A., Albert Einstein: Philosopher-Scientist (1949), The Library of Living Philosophers: The Library of Living Philosophers Evanston, Ill.), 200-241 · Zbl 0038.14804 |
| [5] | Khrennikov, A., Bohr against Bell: complementarity versus nonlocality, Open Phys., 15, 1, 734-738 (2017) |
| [6] | Jaeger, G., Quantum contextuality in the Copenhagen approach, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci., 377, 2157, Article 20190025 pp. (2019) · Zbl 1462.81018 |
| [7] | Khrennikov, A., Interpretations of Probability (2009), Walter de Gruyter: Walter de Gruyter Berlin, New York · Zbl 1369.81014 |
| [8] | Khrennikov, A., Contextual Approach to Quantum Formalism, Fundamental Theories of Physics, vol. 160 (2009), Springer Science + Business Media B.V. · Zbl 1176.81001 |
| [9] | London, F.; Bauer, E., La theorie de l’observation en mécanique quantique; No. 775 of Actualités scientifiques et industrielles: Exposés de physique générale, publiés sous la direction de Paul Langevin (1939), Hermann: Hermann Paris, English translation in [10] · JFM 65.1526.06 |
| [10] | London, F.; Bauer, E., The theory of observation in quantum mechanics, (Wheeler, J. A.; Zurek, W. H., Quantum Theory and Measurement (1983), Princeton University Press: Princeton University Press Princeton, NJ), 217-259, consolidated translation of French original [9] |
| [11] | Zeilinger, A., A foundational principle for quantum mechanics, Found. Phys., 29, 4, 631-643 (1999) |
| [12] | Bell, J. S., On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys., 38, 447-452 (1966) · Zbl 0152.23605 |
| [13] | Hertz, H., Prinzipien der Mechanik (1894), Johann Ambrosius Barth (Arthur Meiner): Johann Ambrosius Barth (Arthur Meiner) Leipzig, mit einem Vorwort von H. von Helmholtz · JFM 67.0970.03 |
| [14] | Specker, E., Die Logik nicht gleichzeitig entscheidbarer Aussagen, Dialectica, 14, 2-3, 239-246 (1960), English translation at |
| [15] | Abbott, A. A.; Calude, C. S.; Svozil, K., A variant of the Kochen-Specker theorem localising value indefiniteness, J. Math. Phys., 56, 10, Article 102201 pp. (2015) · Zbl 1333.81017 |
| [16] | Budroni, C.; Cabello, A.; Gühne, O.; Kleinmann, M.; Larsson, J.-A., Quantum contextuality (Feb. 2021) |
| [17] | Cabello, A., Experimentally testable state-independent quantum contextuality, Phys. Rev. Lett., 101, 21, Article 210401 pp. (2008) |
| [18] | Cabello, A.; Portillo, J. R.; Solís, A.; Svozil, K., Minimal true-implies-false and true-implies-true sets of propositions in noncontextual hidden-variable theories, Phys. Rev. A, 98, Article 012106 pp. (2018) |
| [19] | Gleason, A. M., Measures on the closed subspaces of a Hilbert space, J. Math. Mech., 6, 4, 885-893 (1957) · Zbl 0078.28803 |
| [20] | Zierler, N.; Schlessinger, M., Boolean embeddings of orthomodular sets and quantum logic, Duke Math. J., 32, 251-262 (1965), reprinted in Ref. [78] · Zbl 0171.25403 |
| [21] | Kamber, F., Zweiwertige Wahrscheinlichkeitsfunktionen auf orthokomplementären Verbänden, Math. Ann., 158, 3, 158-196 (1965) · Zbl 0152.46205 |
| [22] | Halmos, P. R., Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics (1958), Springer: Springer New York · Zbl 0107.01404 |
| [23] | Svozil, K., Logical equivalence between generalized urn models and finite automata, Int. J. Theor. Phys., 44, 745-754 (2005) · Zbl 1104.81033 |
| [24] | Wright, R., Generalized urn models, Found. Phys., 20, 7, 881-903 (1990) |
| [25] | Moore, E. F., Gedanken-experiments on sequential machines, (Shannon, C. E.; McCarthy, J., Automata Studies. (AM-34) (1956), Princeton University Press: Princeton University Press Princeton, NJ), 129-153 · Zbl 0074.38401 |
| [26] | Schaller, M.; Svozil, K., Automaton logic, Int. J. Theor. Phys., 35, 911-940 (1996) · Zbl 0854.03041 |
| [27] | Svozil, K., Faithful orthogonal representations of graphs from partition logics, Soft Comput., 24, 10239-10245 (2020) · Zbl 1492.81024 |
| [28] | Froissart, M., Constructive generalization of Bell’s inequalities, Il Nuovo Cimento B (11, 1971-1996), 64, 2, 241-251 (1981) |
| [29] | Pitowsky, I., The range of quantum probability, J. Math. Phys., 27, 6, 1556-1565 (1986) |
| [30] | Foulis, D. J.; Randall, C. H., Empirical logic and quantum mechanics, (Suppes, P., Logic and Probability in Quantum Mechanics (1976), Springer Netherlands: Springer Netherlands Dordrecht), 73-103 · Zbl 0359.02018 |
| [31] | Birkhoff, G.; von Neumann, J., The logic of quantum mechanics, Ann. Math., 37, 4, 823-843 (1936) · JFM 62.1061.04 |
| [32] | Suppes, P.; Zanotti, M., When are probabilistic explanations possible?, Synthese, 48, 2, 191-199 (1981) · Zbl 0476.03011 |
| [33] | Avis, D.; Imai, H.; Ito, T.; Sasaki, Y., Two-party Bell inequalities derived from combinatorics via triangular elimination, J. Phys. A, Math. Gen., 38, 50, 10971-10987 (2005) · Zbl 1092.81508 |
| [34] | Kochen, S.; Specker, E. P., Logical structures arising in quantum theory, (Addison, J. W.; Henkin, L.; Tarski, A., The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley (1965), North Holland: North Holland Amsterdam, New York, Oxford), 177-189, reprinted in Ref. [76, pp. 209-221] · Zbl 0171.25402 |
| [35] | Svozil, K., On generalized probabilities: correlation polytopes for automaton logic and generalized urn models, extensions of quantum mechanics and parameter cheats (2001) |
| [36] | Klyachko, A. A.; Can, M. A.; Binicioğlu, S.; Shumovsky, A. S., Simple test for hidden variables in spin-1 systems, Phys. Rev. Lett., 101, Article 020403 pp. (2008) · Zbl 1228.81068 |
| [37] | Bub, J.; Stairs, A., Contextuality and nonlocality in ‘no signaling’ theories, Found. Phys., 39, 690-711 (2009) · Zbl 1176.81006 |
| [38] | Svozil, K., Quantum violation of the Suppes-Zanotti inequalities and “contextuality”, Int. J. Theor. Phys., 60, 6, 2300-2310 (2021) · Zbl 1528.81030 |
| [39] | Clauser, J. F.; Horne, M. A.; Shimony, A.; Holt, R. A., Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett., 23, 15, 880-884 (1969) · Zbl 1371.81014 |
| [40] | Peres, A., Unperformed experiments have no results, Am. J. Phys., 46, 745-747 (1978) |
| [41] | Svozil, K., Quantum value indefiniteness, Nat. Comput., 10, 4, 1371-1382 (2011) · Zbl 1251.81030 |
| [42] | Svozil, K., How much contextuality?, Nat. Comput., 11, 2, 261-265 (2012) · Zbl 1339.81013 |
| [43] | Dzhafarov, E. N.; Kujala, J. V.; Larsson, J.-A.k., Contextuality in three types of quantum-mechanical systems, Found. Phys., 45, 7, 762-782 (2015) · Zbl 1327.81023 |
| [44] | Dzhafarov, E. N.; Cervantes, V. H.; Kujala, J. V., Contextuality in canonical systems of random variables, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci., 375, 2106, Article 20160389 pp. (2017) · Zbl 1404.81024 |
| [45] | Kujala, J. V.; Dzhafarov, E. N., Measures of contextuality and non-contextuality, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci., 377, 2157, Article 20190149 pp. (2019) · Zbl 1462.81019 |
| [46] | Dzhafarov, E. N.; Kujala, J. V.; Cervantes, V. H., Contextuality and noncontextuality measures and generalized bell inequalities for cyclic systems, Phys. Rev. A, 101, Article 042119 pp. (2020) |
| [47] | Einstein, A.; Podolsky, B.; Rosen, N., Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev., 47, 10, 777-780 (1935) · Zbl 0012.04201 |
| [48] | Garg, A.; Mermin, D. N., Farkas’s lemma and the nature of reality: statistical implications of quantum correlations, Found. Phys., 14, 1, 1-39 (1984) |
| [49] | Ziegler, G. M., Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152 (1994), Springer: Springer New York |
| [50] | Schrijver, A., Theory of Linear and Integer Programming, Wiley Series in Discrete Mathematics & Optimization (1998), John Wiley & Sons: John Wiley & Sons New York, Toronto, London · Zbl 0970.90052 |
| [51] | Fukuda, K., Frequently asked questions in polyhedral computation (2014) |
| [52] | Brody, T. A., The Suppes-Zanotti theorem and the Bell inequalities, Rev. Mex. Fis., 35, 2, 170-187 (1989) · Zbl 1291.81013 |
| [53] | Khrennikov, A., Can there be given any meaning to contextuality without incompatibility?, Int. J. Theor. Phys. (Dec. 2020) |
| [54] | Filipp, S.; Svozil, K., Generalizing Tsirelson’s bound on Bell inequalities using a min-max principle, Phys. Rev. Lett., 93, Article 130407 pp. (2004) |
| [55] | Svozil, K., Extensions of Hardy-type true-implies-false gadgets to classically obtain indistinguishability, Phys. Rev. A, 103, Article 022204 pp. (2021) |
| [56] | Ramanathan, R.; Rosicka, M.; Horodecki, K.; Pironio, S.; Horodecki, M.; Horodecki, P., Gadget structures in proofs of the Kochen-Specker theorem (Aug. 2020) |
| [57] | Svozil, K., New forms of quantum value indefiniteness suggest that incompatible views on contexts are epistemic, Entropy, 20, 6, Article 406 pp. (2018) |
| [58] | Lovász, L., On the Shannon capacity of a graph, IEEE Trans. Inf. Theory, 25, 1, 1-7 (1979) · Zbl 0395.94021 |
| [59] | Belinfante, F. J., A Survey of Hidden-Variables Theories, International Series of Monographs in Natural Philosophy, vol. 55 (1973), Pergamon Press, Elsevier: Pergamon Press, Elsevier Oxford, New York |
| [60] | Redhead, M., Incompleteness, Nonlocality, and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics (1987), Clarendon Press: Clarendon Press Oxford · Zbl 0611.03003 |
| [61] | Cabello, A., A simple proof of the Kochen-Specker theorem, Eur. J. Phys., 15, 4, 179-183 (1994) |
| [62] | Cabello, A., Converting contextuality into nonlocality, Phys. Rev. Lett., 127, Article 070401 pp. (2021) |
| [63] | Tkadlec, J., Greechie diagrams of small quantum logics with small state spaces, Int. J. Theor. Phys., 37, 1, 203-209 (1998) · Zbl 0906.03064 |
| [64] | Svozil, K., Roots and (re)sources of value (in)definiteness versus contextuality, (Hemmo, M.; Shenker, O., Quantum, Probability, Logic: The Work and Influence of Itamar Pitowsky. Quantum, Probability, Logic: The Work and Influence of Itamar Pitowsky, Jerusalem Studies in Philosophy and History of Science (JSPS), vol. 1 (2020), Springer International Publishing: Springer International Publishing Cham), 521-544 · Zbl 1498.81041 |
| [65] | Pitowsky, I., Infinite and finite Gleason’s theorems and the logic of indeterminacy, J. Math. Phys., 39, 1, 218-228 (1998) · Zbl 0921.03057 |
| [66] | Hrushovski, E.; Pitowsky, I., Generalizations of Kochen and Specker’s theorem and the effectiveness of Gleason’s theorem, Stud. Hist. Philos. Sci. Part B, Stud. Hist. Philos. Mod. Phys., 35, 2, 177-194 (2004) · Zbl 1222.81068 |
| [67] | Abbott, A. A.; Calude, C. S.; Conder, J.; Svozil, K., Strong Kochen-Specker theorem and incomputability of quantum randomness, Phys. Rev. A, 86, Article 062109 pp. (2012) |
| [68] | Abbott, A. A.; Calude, C. S.; Svozil, K., Value-indefinite observables are almost everywhere, Phys. Rev. A, 89, Article 032109 pp. (2014) |
| [69] | Jaynes, E. T., Probability Theory: The Logic of Science (2012), Cambridge University Press: Cambridge University Press Cambridge, edited by G. Larry Bretthorst · Zbl 1045.62001 |
| [70] | Kleene, S. C., General recursive functions of natural numbers, Math. Ann., 112, 1, 727-742 (1936) · Zbl 0014.19402 |
| [71] | Abbott, A. A.; Calude, C. S.; Svozil, K., A quantum random number generator certified by value indefiniteness, Math. Struct. Comput. Sci., 24 (2014) · Zbl 1342.65005 |
| [72] | Svozil, K., “Haunted” quantum contextuality (1999) · Zbl 1353.81009 |
| [73] | Svozil, K., Proposed direct test of a certain type of noncontextuality in quantum mechanics, Phys. Rev. A, 80, 4, Article 040102 pp. (2009) |
| [74] | Griffiths, R. B., What quantum measurements measure, Phys. Rev. A, 96, 3 (Sep. 2017) |
| [75] | Griffiths, R. B., Quantum measurements and contextuality, Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci., 377, 2157, Article 20190033 pp. (2019) · Zbl 1462.81021 |
| [76] | Specker, E., Selecta (1990), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0699.01038 |
| [77] | Stigler, S. M.; Gieryn, Thomas F., Stigler’s law of eponymy, Science and Social Structure: a Festschrift for Robert K. Merton. Science and Social Structure: a Festschrift for Robert K. Merton, Trans. N. Y. Acad. Sci., 39, 1 Series II, 147-157 (1980), reprinted in [79] |
| [78] | Zierler, N.; Schlessinger, M., Boolean embeddings of orthomodular sets and quantum logic, (Hooker, C. A., The Logico-Algebraic Approach to Quantum Mechanics: Volume I: Historical Evolution (1975), Springer Netherlands: Springer Netherlands Dordrecht), 247-262 |
| [79] | Stigler, S. M., Statistics on the Table. The History of Statistical Concepts and Methods, 277-290 (2002), Harvard University Press: Harvard University Press Cambridge, MA, USA and London, England |
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