Fine's theorem, noncontextuality, and correlations in Specker's scenario

Abstract

A characterization of noncontextual models which fall within the ambit of Fine's theorem is provided. In particular, the equivalence between the existence of three notions is made explicit: a joint probability distribution over the outcomes of all the measurements considered, a measurement-noncontextual and outcome-deterministic (or KS-noncontextual, where 'KS' stands for 'Kochen-Specker') model for these measurements, and a measurement-noncontextual and factorizable model for them. A KS-inequality, therefore, is implied by each of these three notions. Following this characterization of noncontextual models that fall within the ambit of Fine's theorem, non-factorizable noncontextual models which lie outside the domain of Fine's theorem are considered. While outcome determinism for projective (sharp) measurements in quantum theory can be shown to follow from the assumption of preparation noncontextuality, such a justification is not available for nonprojective (unsharp) measurements which ought to admit outcome-indeterministic response functions. The Liang-Spekkens-Wiseman (LSW) inequality is cited as an example of a noncontextuality inequality that should hold in any noncontextual model of quantum theory without assuming factorizability. Three other noncontextuality inequalities, which turn out to be equivalent to the LSW inequality under relabellings of measurement outcomes, are derived for Specker's scenario. The polytope of correlations admissible in this scenario, given the no-disturbance condition, is characterized.

arXiv comment11 pages, comments welcome. Version 2 is essentially the published version

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Cited by in corpus (12)

• Kochen-Specker Contextuality
• All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed sets of operational equivalences
• Identifying Nonconvexity in the Sets of Limited-Dimension Quantum Correlations
• From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities
• Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements
• Macroscopic non-contextuality as a principle for Almost Quantum Correlations
• Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)
• Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem
• Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem
• Shadows and subsystems of generalized probabilistic theories: when tomographic incompleteness is not a loophole for contextuality proofs
• Connecting extended Wigner's friend arguments and noncontextuality
• Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory