What is Nonclassical about Uncertainty Relations?

Lorenzo Catani, Matthew Leifer, Giovanni Scala, David Schmid, and Robert W. Spekkens
Phys. Rev. Lett. 129, 240401 – Published 9 December 2022
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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.

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  • Received 12 August 2022
  • Accepted 28 October 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.240401

© 2022 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Lorenzo Catani1,*, Matthew Leifer2, Giovanni Scala3, David Schmid3, and Robert W. Spekkens4

  • 1Electrical Engineering and Computer Science Department, Technische Universität Berlin, 10587 Berlin, Germany
  • 2Institute for Quantum Studies and Schmid College of Science and Technology, Chapman University, One University Drive, Orange, California, 92866, USA
  • 3International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdansk, Poland
  • 4Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5

  • *lorenzo.catani@tu-berlin.de

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Issue

Vol. 129, Iss. 24 — 9 December 2022

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