  Uncertainty relations express limits on the extent to which the outcomes of
distinct measurements on a single state can be made jointly predictable. The
existence of nontrivial uncertainty relations in quantum theory is generally
considered to be a way in which it entails a departure from the classical
worldview. However, this perspective is undermined by the fact that there exist
operational theories which exhibit nontrivial uncertainty relations but which
are consistent with the classical worldview insofar as they admit of a
generalized-noncontextual ontological model. This prompts the question of what
aspects of uncertainty relations, if any, cannot be realized in this way and so
constitute evidence of genuine nonclassicality. We here consider uncertainty
relations describing the tradeoff between the predictability of a pair of
binary-outcome measurements (e.g., measurements of Pauli X and Pauli Z
observables in quantum theory). We show that, for a class of theories
satisfying a particular symmetry property, the functional form of this
predictability tradeoff is constrained by noncontextuality to be below a linear
curve. Because qubit quantum theory has the relevant symmetry property, the
fact that its predictability tradeoff describes a section of a circle is a
violation of this noncontextual bound, and therefore constitutes an example of
how the functional form of an uncertainty relation can witness contextuality.
We also deduce the implications for a selected group of operational foils to
quantum theory and consider the generalization to three measurements.

Quantum Physics