  When a measurement is compatible with each of two other measurements that are
incompatible with one another, these define distinct contexts for the given
measurement. The Kochen-Specker theorem rules out models of quantum theory that
satisfy a particular assumption of context-independence: that sharp
measurements are assigned outcomes both deterministically and independently of
their context. This notion of noncontextuality is not suited to a direct
experimental test because realistic measurements always have some degree of
unsharpness due to noise. However, a generalized notion of noncontextuality has
been proposed that is applicable to any experimental procedure, including
unsharp measurements, but also preparations as well, and for which a quantum
no-go result still holds. According to this notion, the model need only specify
a probability distribution over the outcomes of a measurement in a
context-independent way, rather than specifying a particular outcome. It also
implies novel constraints of context-independence for the representation of
preparations. In this article, we describe a general technique for translating
proofs of the Kochen-Specker theorem into inequality constraints on realistic
experimental statistics, the violation of which witnesses the impossibility of
a noncontextual model. We focus on algebraic state-independent proofs, using
the Peres-Mermin square as our illustrative example. Our technique yields the
necessary 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 how
experimental data must be processed in order to achieve a test of these
inequalities. We also provide a criticism of prior proposals for experimental
tests of noncontextuality based on the Peres-Mermin square.
