Quantifiers and witnesses for the nonclassicality of measurements and of states

Yujie Zhang; Yìlè Yīng; David Schmid

doi:10.22331/q-2026-07-30-2180

Quantifiers and witnesses for the nonclassicality of measurements and of states

Yujie Zhang^1,2,3, Yìlè Yīng^3,2, and David Schmid³

¹Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario Canada N2L 3G1
²Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
³Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5

Published:	2026-07-30, volume 10, page 2180
Editor:	Paul Skrzypczyk
Eprint:	arXiv:2504.02944v3
Doi:	https://doi.org/10.22331/q-2026-07-30-2180
Citation:	Quantum 10, 2180 (2026).

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Abstract

In recent work [Phys. Rev. X 16, 021050], we proposed a unified notion of nonclassicality that applies to arbitrary processes in quantum theory, including individual quantum states, measurements, and sets thereof. This notion is derived from the principle of generalized noncontextuality, but in a novel manner that applies to individual processes rather than full experiments or theories. In the present work, we develop semidefinite-programming-based certificates and witnesses for the nonclassicality of states, sources, measurements, and sets thereof. These theory-dependent methods complement theory-independent approaches based on noncontextuality inequalities. We demonstrate the framework through a variety of explicit examples.

Featured image: Nonclassicality of sources, multi-states, multi-sources, measurements, and multi-measurements.

Popular summary

Quantum theory contains many different signatures of nonclassical behavior, such as entanglement, incompatible measurements, and contextuality. In earlier work, we proposed a unified way to ask whether an individual quantum process—such as a single measurement or a collection of states—should itself be regarded as nonclassical. Here we turn that conceptual idea into practical tools. We develop semidefinite-programming methods that quantify how robust the nonclassicality of a process is to noise and construct witnesses that can certify it experimentally. We also explain the distinction between tests that rely on a perfect characterization of the devices and theory-independent tests based only on observed statistics. Applying these methods to a range of examples reveals that some measurements and sets of states that would traditionally appear classical are nevertheless nonclassical in this broader sense. Finally, we show how these ideas can be used to certify entanglement even in cases where standard steering or Bell tests fail, thereby extending the range of quantum states whose nonclassicality can be demonstrated experimentally.

► BibTeX data

@article{Zhang2026quantifiers,
  doi = {10.22331/q-2026-07-30-2180},
  url = {https://doi.org/10.22331/q-2026-07-30-2180},
  title = {Quantifiers and witnesses for the nonclassicality of measurements and of states},
  author = {Zhang, Yujie and Yīng, Y{\`{i}}l{\`{e}} and Schmid, David},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2180},
  month = jul,
  year = {2026}
}

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[2] Mao-Sheng Li, Rafael Wagner, and Lin Zhang, "Multistate imaginarity and coherence in qubit systems", Physical Review A 113 1, 012428 (2026).

[3] Vinicius P. Rossi, Beata Zjawin, Roberto D. Baldijão, David Schmid, John H. Selby, and Ana Belén Sainz, "Typicality of Contextuality", arXiv:2510.20722, (2025).

[4] Yujie Zhang, David Schmid, Yìlè Yīng, and Robert W. Spekkens, "Reassessing the Boundary between Classical and Nonclassical for Individual Quantum Processes", Physical Review X 16 2, 021050 (2026).

[5] Yujie Zhang, Jonah Spodek, David Schmid, Carter Reid, Liam J. Morrison, Thomas Jennewein, Kevin J. Resch, and Robert W. Spekkens, "A new paradigm for entanglement certification using noncontextuality inequalities", arXiv:2507.01122, (2025).

[6] Roberto D. Baldijão, Marco Erba, David Schmid, John H. Selby, and Ana Belén Sainz, "Tomographically-nonlocal entanglement", arXiv:2602.16280, (2026).

The above citations are from SAO/NASA ADS (last updated successfully 2026-08-12 20:45:18). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-12 20:44:15).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.