# 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.

### Abstract

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).

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### Cited by

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

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

[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, "Hypergraph Contextuality", Entropy 21 11, 1107 (2019).

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