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

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.

arXiv comment29 pages, 14 figures + 1 table, accepted for publication in Quantum

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• The Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories
• Contextuality without incompatibility
• Contextuality in entanglement-assisted one-shot classical communication
• Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)
• Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem
• Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios
• Joint measurability structures realizable with qubit measurements: incompatibility via marginal surgery
• Shadows and subsystems of generalized probabilistic theories: when tomographic incompleteness is not a loophole for contextuality proofs
• Emergence of noncontextuality under quantum Darwinism
• Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory