  In a noncontextual hidden variable model of quantum theory, hidden variables
determine the outcomes of every measurement in a manner that is independent of
how the measurement is implemented. Using a generalization of this notion to
arbitrary operational theories and to preparation procedures, we demonstrate
that a particular two-party information-processing task, "parity-oblivious
multiplexing," is powered by contextuality in the sense that there is a limit
to how well any theory described by a noncontextual hidden variable model can
perform. This bound constitutes a "noncontextuality inequality" that is
violated by quantum theory. We report an experimental violation of this
inequality in good agreement with the quantum predictions. The experimental
results also provide the first demonstration of 2-to-1 and 3-to-1 quantum
random access codes. Consequently, our experimental results also demonstrate
better-than-classical performance for this task. They also represent the first
demonstration of 2-to-1 and 3-to-1 quantum random access codes.
