Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem

Ravi Kunjwal

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5,
Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050 Brussels, Belgium.

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Kochen-Specker (KS) theorem reveals the inconsistency between quantum theory and any putative underlying model of it satisfying the constraint of KS-noncontextuality. A logical proof of the KS theorem is one that relies only on the compatibility relations amongst a set of projectors (a KS set) to witness this inconsistency. These compatibility relations can be represented by a hypergraph, referred to as a contextuality scenario. Here we consider contextuality scenarios that we term KS-uncolourable, e.g., those which appear in logical proofs of the KS theorem. We introduce a hypergraph framework to obtain noise-robust witnesses of contextuality from such scenarios.
Our approach builds on the results of R. Kunjwal and R. W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015), by providing new insights into the relationship between the structure of a contextuality scenario and the associated noise-robust noncontextuality inequalities that witness contextuality. The present work also forms a necessary counterpart to the framework presented in R. Kunjwal, Quantum 3, 184 (2019), which only applies to KS-colourable contextuality scenarios, i.e., those which do not admit logical proofs of the KS theorem but do admit statistical proofs.
We rely on a single hypergraph invariant, defined in R. Kunjwal, Quantum 3, 184 (2019), that appears in our contextuality witnesses, namely, the weighted max-predictability. The present work can also be viewed as a study of this invariant. Significantly, unlike the case of R. Kunjwal, Quantum 3, 184 (2019), none of the graph invariants from the graph-theoretic framework for KS-contextuality due to Cabello, Severini, and Winter (the ``CSW framework", Phys. Rev. Lett. 112, 040401 (2014)) are relevant for our noise-robust noncontextuality inequalities.

A video recording of a talk on this paper is available form http://pirsa.org/17070059/.

Drawing on previous work — R. Kunjwal and R.W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015) — that outlined a conceptual scheme to go from a logical proof of the Kochen-Specker theorem to noise-robust witnesses of contextuality, this paper carries out the technical development of a hypergraph framework for systematically obtaining these witnesses. It can also be viewed as a study of a hypergraph invariant that is relevant for these noise-robust witnesses of contextuality. To the best of our knowledge, this invariant does not seem to have been previously studied in the literature on hypergraph theory. The hypergraph framework here complements the framework developed in R. Kunjwal, Quantum 3, 184 (2019). While the conceptual basis for the former is R. Kunjwal and R.W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015), the conceptual basis for the latter is R. Kunjwal and R.W. Spekkens, Phys. Rev. A 97, 052110 (2018).

► BibTeX data

► References

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

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

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

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

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

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

[7] 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).

[8] M. F. Pusey, ``Robust preparation noncontextuality inequalities in the simplest scenario", Phys. Rev. A 98, 022112 (2018).

[9] 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).

[10] 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).

[11] 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).

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

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

[14] R. Kunjwal, ``Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements", Quantum 3, 184 (2019).

[15] 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).

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

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

[18] R. W. Spekkens, ``Leibniz's principle of the identity of indiscernibles as a foundational principle for quantum theory", Information-Theoretic Interpretations of Quantum Mechanics (2016), University of Western Ontario, available at https:/​/​www.youtube.com/​watch?v=HWOkjisIxc4.

[19] R. W. Spekkens, ``The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of Einstein", arXiv:1909.04628 [physics.hist-ph] (2019).

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

[21] A. Peres, ``Two simple proofs of the Kochen-Specker theorem", J. Phys. A 24, L175 (1991).

[22] N. D. Mermin, ``Hidden variables and the two theorems of John Bell", Rev. Mod. Phys. 65, 803 (1993).

[23] 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).

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

[25] 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).

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

[27] S. Abramsky, R. S. Barbosa, K. Kishida, R. Lal, and S. Mansfield, ``Possibilities determine the combinatorial structure of probability polytopes", Journal of Mathematical Psychology, Volume 74, Pages 58-65 (2016).

[28] S. Mansfield and R. S. Barbosa, ``Extendability in the sheaf-theoretic approach: Construction of Bell models from Kochen-Specker models", arXiv:1402.4827 [quant-ph] (2014).

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

[30] 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).

[31] A. Cabello, ``Twin inequality for fully contextual quantum correlations", Phys. Rev. A 87, 010104(R) (2013).

[32] P. Lisoněk, P. Badziag, J. R. Portillo, and A. Cabello, ``Kochen-Specker set with seven contexts", Phys. Rev. A 89, 042101 (2014).

[33] M. Pavicic, J-P. Merlet, B. McKay, and N. D. Megill, ``Kochen-Specker vectors", J. Phys. A: Math. Gen. 38 1577 (2005).

[34] S. Mansfield, ``The Mathematical Structure of Non-locality and Contextuality", PhD thesis, University of Oxford (2013).

[35] G. Carù, ``Towards a complete cohomology invariant for non-locality and contextuality", arXiv:1807.04203 [quant-ph] (2018).

[36] A. Krishna, ``Experimentally Testable Noncontextuality Inequalities Via Fourier-Motzkin Elimination", M.Sc. Thesis, University of Waterloo (2015).

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

Cited by

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

[2] 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).

[3] 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).

[4] Mladen Pavičić and Norman Megill, "Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces", Entropy 20 12, 928 (2018).

[5] Ravi Kunjwal, "Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements", arXiv:1709.01098.

[6] Mladen Pavičić, "Hypergraph Contextuality", Entropy 21 11, 1107 (2019).

The above citations are from Crossref's cited-by service (last updated successfully 2021-03-06 09:34:58) and SAO/NASA ADS (last updated successfully 2021-03-06 09:34:59). The list may be incomplete as not all publishers provide suitable and complete citation data.