Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements

Ravi Kunjwal

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5.

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We develop a hypergraph-theoretic framework for Spekkens contextuality applied to Kochen-Specker (KS) type scenarios that goes beyond the Cabello-Severini-Winter (CSW) framework. To do this, we add new hypergraph-theoretic ingredients to the CSW framework. We then obtain noise-robust noncontextuality inequalities in this generalized framework by applying the assumption of (Spekkens) noncontextuality to both preparations and measurements. The resulting framework goes beyond the CSW framework in both senses, conceptual and technical. On the conceptual level: 1) as in any treatment based on the generalized notion of noncontextuality à la Spekkens, we relax the assumption of outcome determinism inherent to the Kochen-Specker theorem but retain measurement noncontextuality, besides introducing preparation noncontextuality, 2) we do not require the $\textit{exclusivity }$ $\textit{principle}$ -- that pairwise exclusive measurement events must all be mutually exclusive -- as a fundamental constraint on measurement events of interest in an experimental test of contextuality, given that this property is not true of general quantum measurements, and 3) as a result, we do not need to presume that measurement events of interest are ``sharp" (for any definition of sharpness), where this notion of sharpness is meant to imply the exclusivity principle. On the technical level, we go beyond the CSW framework in the following senses: 1) we introduce a source events hypergraph -- besides the measurement events hypergraph usually considered -- and define a new operational quantity ${\rm Corr}$ that appears in our inequalities, 2) we define a new hypergraph invariant -- the $\textit{weighted}$ $\textit{max-predictability}$ -- that is necessary for our analysis and appears in our inequalities, and 3) our noise-robust noncontextuality inequalities quantify tradeoff relations between three operational quantities -- ${\rm Corr}$, $R$, and $p_0$ -- only one of which (namely, $R$) corresponds to the Bell-Kochen-Specker functionals appearing in the CSW framework; when ${\rm Corr}=1$, the inequalities formally reduce to CSW type bounds on $R$. Along the way, we also consider in detail the scope of our framework vis-à-vis the CSW framework, particularly the role of Specker's principle in the CSW framework, i.e., what the principle means for an operational theory satisfying it and why we don't impose it in our framework.

We present a hypergraph framework to bridge the gap between two approaches to contextuality, Kochen-Specker contextuality and Spekkens contextuality. In particular, we show how to use witnesses for the former in the Cabello-Severini-Winter framework to obtain witnesses for the latter in our framework. This opens the path to a fruitful exchange of ideas between the two approaches.

► BibTeX data

► References

[1] L. Hardy, ``Quantum Theory From Five Reasonable Axioms", arXiv:quant-ph/​0101012 (2001).

[2] L. Masanes and M. P. Mueller, ``A derivation of quantum theory from physical requirements", New J. Phys. 13, 063001 (2011).

[3] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Probabilistic theories with purification", Phys. Rev. A 81, 062348 (2010).

[4] J. S. Bell, ``On the Einstein-Podolsky-Rosen paradox", Physics 1, 195 (1964). Reprinted in Ref. Bellbook, Chapter 2.

[5] J. S. Bell, ``On the problem of hidden variables in quantum mechanics", Rev. Mod. Phys. 38, 447 (1966). Reprinted in Ref. Bellbook, Chapter 1.

[6] J. S. Bell, ``Speakable and Unspeakable in Quantum Mechanics", 2nd Edition, Cambridge University Press, 2004.

[7] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, ``Proposed Experiment to Test Local Hidden-Variable Theories", Phys. Rev. Lett. 23, 880 (1969).

[8] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, ``Bell nonlocality", Rev. Mod. Phys. 86, 419 (2014).

[9] B. Hensen et al., ``Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres", Nature 526, 682 - 686 (2015).

[10] Lynden K. Shalm et al., ``Strong Loophole-Free Test of Local Realism", Phys. Rev. Lett. 115, 250402 (2015).

[11] M. Giustina et al., ``Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons", Phys. Rev. Lett. 115, 250401 (2015).

[12] R. Kunjwal and R. W. Spekkens, ``From the Kochen-Specker Theorem to Noncontextuality Inequalities without Assuming Determinism", Phys. Rev. Lett. 115, 110403 (2015).

[13] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, R. W. Spekkens, ``An experimental test of noncontextuality without unphysical idealizations", Nat. Commun. 7, 11780 (2016).

[14] A. Krishna, R. W. Spekkens, and E. Wolfe, ``Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen-Specker theorem: the Peres-Mermin square'', New J. Phys 19, 123031 (2017).

[15] D. Schmid and R. W. Spekkens, ``Contextual Advantage for State Discrimination", Phys. Rev. X 8, 011015 (2018).

[16] R. Kunjwal and R. W. Spekkens, ``From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities", Phys. Rev. A 97, 052110 (2018).

[17] D. Schmid, R. W. Spekkens, and E. Wolfe, ``All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences", Phys. Rev. A 97, 062103 (2018).

[18] R. W. Spekkens, ``Contextuality for preparations, transformations, and unsharp measurements'', Phys. Rev. A 71, 052108 (2005).

[19] S. Kochen and E. P. Specker, ``The Problem of Hidden Variables in Quantum Mechanics", J. Math. Mech. 17, 59 (1967). Also available at JSTOR.

[20] N. Harrigan and R. W. Spekkens,``Einstein, Incompleteness, and the Epistemic View of Quantum States,'' Found. Phys. 40, 125 (2010).

[21] A. Cabello, S. Severini, and A. Winter, ``(Non-)Contextuality of Physical Theories as an Axiom", arXiv:1010.2163 [quant-ph] (2010).

[22] A. Cabello, S. Severini, and A. Winter, ``Graph-Theoretic Approach to Quantum Correlations", Phys. Rev. Lett. 112, 040401 (2014).

[23] A. Acín, T. Fritz, A. Leverrier, and A. B. Sainz, A Combinatorial Approach to Nonlocality and Contextuality, Comm. Math. Phys. 334(2), 533-628 (2015).

[24] J. Barrett, ``Information processing in generalized probabilistic theories", Phys. Rev. A 75, 032304 (2007).

[25] S. Abramsky and A. Brandenburger, ``The sheaf-theoretic structure of non-locality and contextuality", New J. Phys. 13, 113036 (2011).

[26] A. Fine, ``Hidden Variables, Joint Probability, and the Bell Inequalities", Phys. Rev. Lett. 48, 291 (1982).

[27] R. Kunjwal, ``Fine's theorem, noncontextuality, and correlations in Specker's scenario", Phys. Rev. A 91, 022108 (2015).

[28] R. W. Spekkens, ``The Status of Determinism in Proofs of the Impossibility of a Noncontextual Model of Quantum Theory", Found. Phys. 44, 1125-1155 (2014).

[29] A. Cabello, ``What do we learn about quantum theory from Kochen-Specker quantum contextuality?", PIRSA:17070034 (2017).

[30] G. Chiribella and X. Yuan, ``Measurement sharpness cuts nonlocality and contextuality in every physical theory", arXiv:1404.3348 [quant-ph] (2014).

[31] G. Chiribella and X. Yuan, ``Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality", Information and Computation, 250, 15-49 (2016).

[32] R. Chaves and T. Fritz, ``Entropic approach to local realism and noncontextuality", Phys. Rev. A 85, 032113 (2012).

[33] Tobias Fritz and Rafael Chaves, ``Entropic Inequalities and Marginal Problems", IEEE Trans. on Information Theory, vol. 59, pages 803 - 817 (2013).

[34] R. Kunjwal, ``Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem", arXiv:1805.02083 [quant-ph] (2018).

[35] A. Cabello, ``Specker's fundamental principle of quantum mechanics", arXiv:1212.1756 [quant-ph] (2012).

[36] R. W. Spekkens, ``Noncontextuality: how we should define it, why it is natural, and what to do about its failure", PIRSA:17070035 (2017).

[37] M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens, ``Experimentally bounding deviations from quantum theory in the landscape of generalized probabilistic theories", arXiv:1710.05948 [quant-ph] (2017).

[38] M. F. Pusey, L. del Rio, and B. Meyer, ``Contextuality without access to a tomographically complete set", arXiv:1904.08699 (2019).

[39] Y. C. Liang, R. W. Spekkens, H. M. Wiseman, ``Specker's parable of the overprotective seer: A road to contextuality, nonlocality and complementarity'', Phys. Rep. 506, 1 (2011).

[40] R. Kunjwal and S. Ghosh, ``Minimal state-dependent proof of measurement contextuality for a qubit", Phys. Rev. A 89, 042118 (2014).

[41] R. Kunjwal, C. Heunen, and T. Fritz, ``Quantum realization of arbitrary joint measurability structures", Phys. Rev. A 89, 052126 (2014).

[42] R. Kunjwal, ``A note on the joint measurability of POVMs and its implications for contextuality", arXiv:1403.0470 [quant-ph] (2014).

[43] S. Popescu and D. Rohrlich, ``Quantum nonlocality as an axiom", Found. Phys. 24, 379-385 (1994).

[44] T. Heinosaari, D. Reitzner, and P. Stano, ``Notes on Joint Measurability of Quantum Observables", Found. Phys. 38, 1133-1147 (2008).

[45] R. Kunjwal, ``How to go from the KS theorem to experimentally testable noncontextuality inequalities", PIRSA:17070059 (2017).

[46] Konrad Engel, ``Sperner theory: Encyclopedia of Mathematics and its Applications", Vol. 65, Cambridge University Press, Cambridge (1997).

[47] A. A. Klyachko, M. A. Can, S. Binicioğlu, and A. S. Shumovsky, ``Simple Test for Hidden Variables in Spin-1 Systems", Phys. Rev. Lett. 101, 020403 (2008).

[48] C. Held, ``The Kochen-Specker Theorem", The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), Edward N. Zalta (ed.).

[49] T. Gonda, R. Kunjwal, D. Schmid, E. Wolfe, and A. B. Sainz, ``Almost Quantum Correlations are Inconsistent with Specker's Principle", Quantum 2, 87 (2018).

[50] M. Navascués, Y. Guryanova, M. J. Hoban, and A. Acín, ``Almost quantum correlations", Nat. Commun. 6, 6288 (2015).

[51] A. Cabello, Adan, J. Estebaranz, and G. Garcia-Alcaine, ``Bell-Kochen-Specker theorem: A proof with 18 vectors,'' Phys. Lett. A 212, 183 (1996).

[52] E. G. Beltrametti and S. Bugajski, ``A classical extension of quantum mechanics", J. Phys. A 28, 3329 (1995).

[53] X. Zhan, E. G. Cavalcanti, J. Li, Z. Bian, Y. Zhang, H. M. Wiseman, and P. Xue, ``Experimental generalized contextuality with single-photon qubits", Optica 4, 966-971 (2017).

[54] R. Kunjwal, ``Contextuality beyond the Kochen-Specker theorem", arXiv:1612.07250 [quant-ph] (2016).

[55] T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, ``Local orthogonality: a multipartite principle for correlations", Nat. Commun. 4, 2263 (2013).

[56] R. W. Spekkens, ``Nonclassicality as the failure of noncontextuality", PIRSA:15050081 (2015) (see the slide at 41:43 minutes).

[57] R. W. Spekkens, ``Quasi-Quantization: Classical Statistical Theories with an Epistemic Restriction", In: Chiribella G., Spekkens R. (eds) Quantum Theory: Informational Foundations and Foils. Fundamental Theories of Physics, vol 181. Springer, Dordrecht.

[58] T. Vidick and S. Wehner, ``Does Ignorance of the Whole Imply Ignorance of the Parts? Large Violations of Noncontextuality in Quantum Theory", Phys. Rev. Lett. 107, 030402 (2011).

[59] R. Raussendorf, ``Contextuality in measurement-based quantum computation", Phys. Rev. A 88, 022322 (2013).

[60] M. Howard, J. Wallman, V. Veitch, and J. Emerson, ``Contextuality supplies the `magic' for quantum computation", Nature 510, 351 (2014).

[61] N. Delfosse, P. A. Guerin, J. Bian, and R. Raussendorf, ``Wigner Function Negativity and Contextuality in Quantum Computation on Rebits", Phys. Rev. X 5, 021003 (2015).

[62] J. Bermejo-Vega, N. Delfosse, D. E. Browne, C. Okay, R. Raussendorf, ``Contextuality as a resource for qubit quantum computation", Phys. Rev. Lett. 119, 120505 (2017).

[63] J. Singh, K. Bharti, and Arvind, ``Quantum key distribution protocol based on contextuality monogamy", Phys. Rev. A 95, 062333 (2017).

[64] A. Cabello, ``Kochen-Specker Theorem for a Single Qubit using Positive Operator-Valued Measures", Phys. Rev. Lett. 90, 190401 (2003).

[65] A. Grudka and P. Kurzyński, ``Is There Contextuality for a Single Qubit?", Phys. Rev. Lett. 100, 160401 (2008).

[66] P. Busch, ``Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem", Phys. Rev. Lett. 91, 120403 (2003).

[67] C. M. Caves, C. A. Fuchs, K. Manne, and J. M. Renes, ``Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements", Found. Phys. 34, 193 (2004).

[68] A. M. Gleason, ``Measures on the closed subspaces of a Hilbert space", J. Math. Mech. 6, 885 (1957). Also available at JSTOR.

[69] P. K. Aravind, ``The generalized Kochen-Specker theorem", Phys. Rev. A 68, 052104 (2003).

[70] A. A. Methot, ``Minimal Bell-Kochen-Specker proofs with POVMs on qubits", Int. J. Quantum Inf. 5, 353 (2007).

[71] Q. Zhang, H. Li, T. Yang, J. Yin, J. Du, J. W. Pan, ``Experimental Test of the Kochen-Specker Theorem for Single Qubits using Positive Operator-Valued Measures", arXiv:quant-ph/​0412049 (2004).

[72] L. Mancinska, G. Scarpa, and S. Severini, ``New Separations in Zero-Error Channel Capacity Through Projective Kochen Specker Sets and Quantum Coloring", IEEE Transactions on Information Theory 59, 4025 (2013).

[73] J. Henson and A. B. Sainz, ``Macroscopic noncontextuality as a principle for almost-quantum correlations", Phys. Rev. A 91, 042114 (2015).

[74] D. A. Meyer, ``Finite Precision Measurement Nullifies the Kochen-Specker Theorem", Phys. Rev. Lett. 83, 3751 (1999).

[75] A. Kent, ``Noncontextual Hidden Variables and Physical Measurements", Phys. Rev. Lett. 83, 3755 (1999).

[76] R. Clifton and A. Kent, ``Simulating quantum mechanics by non-contextual hidden variables", Proc. R. Soc. Lond. A: Vol. 456, 2101-2114 (2000).

[77] J. Barrett and A. Kent, ``Non-contextuality, finite precision measurement and the Kochen-Specker theorem", Stud. Hist. Philos. Mod. Phys. 35, 151 (2004).

[78] A. Winter, ``What does an experimental test of quantum contextuality prove or disprove?", J. Phys. A: Math. Theor. 47, 424031 (2014).

[79] V. B. Scholz and R. F. Werner, ``Tsirelson's Problem", arXiv:0812.4305 [math-ph] (2008).

[80] T. Fritz, ``Tsirelson's problem and Kirchberg's conjecture", Rev. Math. Phys. 24 (5), 1250012 (2012).

Cited by

[1] Ravi Kunjwal, "Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem", Quantum 4, 219 (2020).

[2] Sacha Huriot-Tattegrain and Mehdi Mhalla, "Contextuality and Expressivity of Non-locality", Electronic Proceedings in Theoretical Computer Science 340, 160 (2021).

[3] Victoria J Wright and Ravi Kunjwal, "Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem", Quantum 7, 900 (2023).

[4] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson, "Kochen-Specker contextuality", Reviews of Modern Physics 94 4, 045007 (2022).

[5] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Contextuality without Incompatibility", Physical Review Letters 130 23, 230201 (2023).

[6] David Schmid, Haoxing Du, John H. Selby, and Matthew F. Pusey, "Uniqueness of Noncontextual Models for Stabilizer Subtheories", Physical Review Letters 129 12, 120403 (2022).

[7] Nikola Andrejic and Ravi Kunjwal, "Joint measurability structures realizable with qubit measurements: Incompatibility via marginal surgery", Physical Review Research 2 4, 043147 (2020).

[8] Rafael Wagner, Anita Camillini, and Ernesto F. Galvão, "Coherence and contextuality in a Mach-Zehnder interferometer", Quantum 8, 1240 (2024).

[9] Shiv Akshar Yadavalli and Ravi Kunjwal, "Contextuality in entanglement-assisted one-shot classical communication", Quantum 6, 839 (2022).

[10] David Schmid, John H. Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens, "Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories", PRX Quantum 2 1, 010331 (2021).

[11] Rafael Wagner, Rui Soares Barbosa, and Ernesto F. Galvão, "Inequalities witnessing coherence, nonlocality, and contextuality", Physical Review A 109 3, 032220 (2024).

[12] Anubhav Chaturvedi, Máté Farkas, and Victoria J Wright, "Characterising and bounding the set of quantum behaviours in contextuality scenarios", Quantum 5, 484 (2021).

[13] Rafael Wagner, Roberto D Baldijão, Alisson Tezzin, and Bárbara Amaral, "Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios", Journal of Physics A: Mathematical and Theoretical 56 50, 505303 (2023).

[14] Kim Vallée, Pierre-Emmanuel Emeriau, Boris Bourdoncle, Adel Sohbi, Shane Mansfield, and Damian Markham, "Corrected Bell and non-contextuality inequalities for realistic experiments", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 382 2268, 20230011 (2024).

[15] Roberto D. Baldijão, Rafael Wagner, Cristhiano Duarte, Bárbara Amaral, and Marcelo Terra Cunha, "Emergence of Noncontextuality under Quantum Darwinism", PRX Quantum 2 3, 030351 (2021).

[16] David Schmid, Robert W. Spekkens, and Elie Wolfe, "All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences", Physical Review A 97 6, 062103 (2018).

[17] Andris Ambainis, Manik Banik, Anubhav Chaturvedi, Dmitry Kravchenko, and Ashutosh Rai, "Parity oblivious d-level random access codes and class of noncontextuality inequalities", Quantum Information Processing 18 4, 111 (2019).

[18] Shane Mansfield and Elham Kashefi, "Quantum Advantage from Sequential-Transformation Contextuality", Physical Review Letters 121 23, 230401 (2018).

[19] Ravi Kunjwal and Robert W. Spekkens, "From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities", Physical Review A 97 5, 052110 (2018).

[20] Cristhiano Duarte and Barbara Amaral, "Resource theory of contextuality for arbitrary prepare-and-measure experiments", Journal of Mathematical Physics 59 6, 062202 (2018).

[21] Tomáš Gonda, Ravi Kunjwal, David Schmid, Elie Wolfe, and Ana Belén Sainz, "Almost Quantum Correlations are Inconsistent with Specker's Principle", Quantum 2, 87 (2018).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-19 08:12:22) and SAO/NASA ADS (last updated successfully 2024-04-19 08:12:23). The list may be incomplete as not all publishers provide suitable and complete citation data.