What is nonclassical about uncertainty relations?

TL;DR

This paper investigates the nonclassical nature of quantum uncertainty relations by analyzing their functional forms and their relation to contextuality, showing that quantum tradeoffs violate classical bounds and thus reveal genuine nonclassical features.

Contribution

It demonstrates that the shape of uncertainty relations in quantum theory witnesses contextuality, providing a new perspective on what makes quantum uncertainty fundamentally nonclassical.

Findings

Quantum tradeoff functions can violate noncontextual bounds.

Quantum uncertainty relations can serve as witnesses of contextuality.

The analysis extends to three-measurement scenarios.

Abstract

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.