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

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.