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Commutativity, Comeasurability, and Contextuality in the Kochen-Specker Arguments

Published online by Cambridge University Press:  01 January 2022

Gábor Hofer-Szabó
Gábor Hofer-Szabó*
Affiliation:
To contact the author, please write to: Research Center for the Humanities, Budapest; e-mail: szabo.gabor@btk.mta.hu.
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Abstract

I will argue that Kochen-Specker arguments do not provide an algebraic proof for quantum contextuality since, for the argument to be effective, (1) operators must be uniquely associated with measurements and (2) commuting operators must represent simultaneous measurements. However, in all Kochen-Specker arguments discussed in the literature either 1 or 2 is not met. Arguments meeting 1 contain commuting operators that do not represent simultaneous measurements and hence fail to physically justify the functional composition principle. Arguments meeting 2 associate some operators with more than one measurement and hence need to invoke an extra assumption different from noncontextuality.

Type
Research Article
Information
Philosophy of Science , Volume 88 , Issue 3 , July 2021 , pp. 483 - 510
Copyright
Copyright 2021 by the Philosophy of Science Association. All rights reserved.

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Footnotes

†

This work has been supported by the Hungarian Scientific Research Fund (K-115593 and K-134275) and a Senior Research Scholarship of the Institute of Advanced Studies Kőszeg. I wish to thank the members of the Budapest Research Group on the Philosophical Foundations of Science, especially Márton Gömöri and Balázs Gyenis for valuable discussions and Karim Thebault for reading the final version of the article.

References

Abramsky, Samson, and Brandenburger, Adam. 2011. “The Sheaf-Theoretic Structure of Non-locality and Contextuality.” New Journal of Physics 13:113036.10.1088/1367-2630/13/11/113036CrossRefGoogle Scholar
Acín, Antonio, Fritz, Tobias, Leverrier, Anthony, and Sainz, Ana Belén. 2015. “A Combinatorial Approach to Nonlocality and Contextuality.” Communications in Mathematical Physics 334 (2): 533–628..CrossRefGoogle Scholar
Barrett, Johathan, and Kent, Adrian. 2004. “Non-contextuality, Finite Precision Measurement and the Kochen-Specker Theorem.” Studies in History and Philosophy of Modern Physics 35:151–76.CrossRefGoogle Scholar
Bell, John S. 1966/2004. “On the Problem of Hidden Variables in Quantum Mechanics.” Repr. in Speakable and Unspeakable in Quantum Mechanics. Cambridge: Cambridge University Press.Google Scholar
Cabello, Adán. 1997. “A Proof with 18 Vectors of the Bell-Kochen-Specker Theorem.” In New Developments on Fundamental Problems in Quantum Physics. ed. Melgar, Miguel Ferrero and Van Der Merwe, Alwyn, 59–62. Dordrecht: Kluwer.CrossRefGoogle Scholar
Cabello, Adán, Severini, Simone, and Winter, Andreas. 2014. “Graph-Theoretic Approach to Quantum Correlations.” Physical Review Letters 112 (4): 040401.CrossRefGoogle ScholarPubMed
Clifton, Robert, and Kent, Adrian. 2000. “Simulating Quantum Mechanics by Non-contextual Hidden Variables.” Proceedings of the Royal Society A 456:2101–14.CrossRefGoogle Scholar
Greenberger, Daniel M., Horne, Michael A., and Zeilinger, Anton. 1989. “Going beyond Bell’s Theorem.” In Bell’s Theorem, Quantum Theory and Conceptions of the Universe, ed. Kafatos, Menas, 69–72. Dordrecht: Kluwer.CrossRefGoogle Scholar
Halmos, Paul R. 1958. Finite-Dimensional Vector Spaces. Dordrecht: Springer.Google Scholar
Held, Carsten. 2018. “The Kochen-Specker Theorem.” In Stanford Encyclopedia of Philosophy, ed. Zalta, Edward N.. Stanford, CA: Stanford University. .Google Scholar
Hermens, Ronnie. 2011. “The Problem of Contextuality and the Impossibility of Experimental Metaphysics Thereof.” Studies in History and Philosophy of Modern Physics 42 (4): 214–25..CrossRefGoogle Scholar
Hofer-Szabó, Gábor. 2021a. “Sequential Measurements Are Not Simultaneous Measurements.” Unpublished manuscript.Google Scholar
Hofer-Szabó, Gábor. 2021b. “Three Noncontextual Hidden Variable Models for the Peres-Mermin Square.” European Journal for the Philosophy of Science 11 (30), forthcoming.CrossRefGoogle Scholar
Hofer-Szabó, Gábor. 2021c. “Two Concepts of Noncontextuality in Quantum Mechanics.” Unpublished manuscript.Google Scholar
Klyachko, Alexander A., Can, M. Ali, Binicioğlu, Sinem, and Shumovsky, Alexander S.. 2008. “A Simple Test for Hidden Variables in Spin-1 Systems.” Physical Review Letters 101:020403.CrossRefGoogle ScholarPubMed
Kochen, Simon, and Specker, Ernst P.. 1967. “The Problem of Hidden Variables in Quantum Mechanics.” Journal of Mathematics and Mechanics 17:59–87.Google Scholar
Krishna, Anirudh, Spekkens, Robert W., and Wolfe, Elie. 2017. “Deriving Robust Noncontextuality Inequalities from Algebraic Proofs of the Kochen-Specker Theorem: The Peres-Mermin Square.” New Journal of Physics 19:123031.CrossRefGoogle Scholar
Lapkiewicz, Radek, Li, Peizhe, Schaeff, Christoph, Langford, Nathan K., Ramelow, Sven, Wieśniak, Marcin, and Zeilinger, Anton. 2011. “Experimental Non-classicality of an Indivisible Quantum System.” Nature 474:490–93.CrossRefGoogle ScholarPubMed
Larsson, Jan-Ake. 2002. “A Kochen-Specker Inequality.” Europhysics Letters 58:799–805.CrossRefGoogle Scholar
Leifer, Matt. 2014. “Is the Quantum State Real? An Extended Review of ψ-Ontology Theorems.” Quanta 3 (1): 67–155..CrossRefGoogle Scholar
Liang, Yeong-Cherng, Spekkens, Robert W., and Wisemand, Howard M.. 2011. “Specker’s Parable of the Overprotective Seer: A Road to Contextuality Nonlocality and Complementarity.” Physics Reports 506 (1–2): 1–39.CrossRefGoogle Scholar
Maroney, Owen J. E., and Timpson, Christopher G.. 2014. “Quantum- vs. Macro-Realism: What Does the Leggett-Garg Inequality Actually Test?” .Google Scholar
Mazurek, Michael D., Pusey, Matthew F., Kunjwal, Ravi, Resch, Kevin J., and Spekkens, Robert W.. 2016. “An Experimental Test of Noncontextuality without Unwarranted Idealizations.” Nature Communications 7:11780.CrossRefGoogle Scholar
Mermin, David. 1993. “Ontological States and the Two Theorems of John Bell.” Reviews of Modern Physics 65 (3): 803–15..CrossRefGoogle Scholar
Meyer, David A. 1999. “Finite Precision Measurement Nullifies the Kochen-Specker Theorem.” Physical Review Letters 83:3751–54.CrossRefGoogle Scholar
Park, James L., and Margenau, Henry. 1968. “Simultaneous Measurability in Quantum Theory.” International Journal of Theoretical Physics 1:211–83.CrossRefGoogle Scholar
Peres, Asher. 1990. “Incompatible Results of Quantum Measurements.” Physics Letters A 151:107–8.Google Scholar
Reck, Michael, Zeilinger, Anton, Bernstein, Herbert J., and Bertani, Philip. 1994. “Experimental Realization of Any Discrete Unitary Operator.” Physical Review Letters 73 (1): 58–91..CrossRefGoogle ScholarPubMed
Redhead, Michael. 1989. Incompleteness, Nonlocality and Realism. Oxford: Oxford University Press.Google Scholar
Shimony, Abner. 1986. “Events and Processes in the Quantum World.” In Quantum Concepts in Space and Time, ed. Penrose, Roger and Isham, C. J., 182–203. Oxford: Clarendon.Google Scholar
Simon, Christoph, Brukner, Časlav, and Zeilinger, Anton. 2001. “Hidden Variable Theorems for Real Experiments.” Physical Review Letters 86:4427–30.CrossRefGoogle ScholarPubMed
Spekkens, Robert W. 2005. “Contextuality for Preparations, Transformations, and Unsharp Measurements.” Physical Review A 71:052108.CrossRefGoogle Scholar
van Fraassen, Bas C. 1979. “Hidden Variables and the Modal Interpretation of Quantum Theory.” Synthese 42:155–65.CrossRefGoogle Scholar