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/.
 J. Barrett and A. Kent, ``Non-contextuality, finite precision measurement and the Kochen-Specker theorem", Stud. Hist. Philos. Mod. Phys. 35, 151 (2004).
 R. Kunjwal and R. W. Spekkens, ``From the Kochen-Specker Theorem to Noncontextuality Inequalities without Assuming Determinism", Phys. Rev. Lett. 115, 110403 (2015).
 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).
 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).
 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).
 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).
 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).
 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.
 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).
 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).
 A. Cabello, S. Severini, and A. Winter, ``Graph-Theoretic Approach to Quantum Correlations", Phys. Rev. Lett. 112, 040401 (2014).
 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).
 S. Abramsky and A. Brandenburger, ``The sheaf-theoretic structure of non-locality and contextuality", New J. Phys. 13, 113036 (2011).
 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).
 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).
 S. Mansfield, ``The Mathematical Structure of Non-locality and Contextuality", PhD thesis, University of Oxford (2013).
 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).
 Nikola Andrejic and Ravi Kunjwal, "Joint measurability structures realizable with qubit measurements: Incompatibility via marginal surgery", Physical Review Research 2 4, 043147 (2020).
 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).
 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).
 Ravi Kunjwal, "Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements", arXiv:1709.01098.
 Mladen Pavičić and Norman Megill, "Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces", Entropy 20 12, 928 (2018).
 Mladen Pavičić, "Hypergraph Contextuality", Entropy 21 11, 1107 (2019).
The above citations are from Crossref's cited-by service (last updated successfully 2022-01-25 07:25:58) and SAO/NASA ADS (last updated successfully 2022-01-25 07:25:59). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.