  An operational definition of contextuality is introduced which generalizes
the standard notion in three ways: (1) it applies to arbitrary operational
theories rather than just quantum theory, (2) it applies to arbitrary
experimental procedures, rather than just sharp measurements, and (3) it
applies to a broad class of ontological models of quantum theory, rather than
just deterministic hidden variable models. We derive three no-go theorems for
ontological models, each based on an assumption of noncontextuality for a
different sort of experimental procedure; one for preparation procedures,
another for unsharp measurement procedures (that is, measurement procedures
associated with positive-operator valued measures), and a third for
transformation procedures. All three proofs apply to two-dimensional Hilbert
spaces, and are therefore stronger than traditional proofs of contextuality.
