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When a measurement is compatible with each of two other measurements that areincompatible with one another, these define distinct contexts for the givenmeasurement. The Kochen-Specker theorem rules out models of quantum theory thatsatisfy a particular assumption of context-independence: that sharpmeasurements are assigned outcomes both deterministically and independently oftheir context. This notion of noncontextuality is not suited to a directexperimental test because realistic measurements always have some degree ofunsharpness due to noise. However, a generalized notion of noncontextuality hasbeen proposed that is applicable to any experimental procedure, includingunsharp measurements, but also preparations as well, and for which a quantumno-go result still holds. According to this notion, the model need only specifya probability distribution over the outcomes of a measurement in acontext-independent way, rather than specifying a particular outcome. It alsoimplies novel constraints of context-independence for the representation ofpreparations. In this article, we describe a general technique for translatingproofs of the Kochen-Specker theorem into inequality constraints on realisticexperimental statistics, the violation of which witnesses the impossibility ofa noncontextual model. We focus on algebraic state-independent proofs, usingthe Peres-Mermin square as our illustrative example. Our technique yields thenecessary and sufficient conditions for a particular set of correlations(between the preparations and the measurements) to admit a noncontextual model.The inequalities thus derived are demonstrably robust to noise. We specify howexperimental data must be processed in order to achieve a test of theseinequalities. We also provide a criticism of prior proposals for experimentaltests of noncontextuality based on the Peres-Mermin square.

Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen-Specker theorem: the Peres-Mermin square