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Quantifiers and witnesses for the nonclassicality of measurements and of states

Yujie Zhang, Yìlè Yīng, and David Schmid

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AbstractIn 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.Popular summaryQuantum 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} }► References [1] Yujie Zhang, David Schmid, Yìlè Yīng, and Robert W. Spekkens. ``Reassessing the boundary between classical and nonclassical for individual quantum processes''. Phys. Rev. X 16, 021050 (2026). https://doi.org/10.1103/vqfz-wzjg [2] R. W. 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Rev. A 75, 032110 (2007). https://doi.org/10.1103/PhysRevA.75.032110 [97] John H. Selby, Elie Wolfe, David Schmid, Ana Belén Sainz, and Vinicius P. Rossi. ``Linear program for testing nonclassicality and an open-source implementation''. Phys. Rev. Lett. 132, 050202 (2024). https://doi.org/10.1103/PhysRevLett.132.050202Cited byCould not fetch Crossref cited-by data during last attempt 2026-07-30 08:22:28: Could not fetch cited-by data for 10.22331/q-2026-07-30-2180 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-07-30 08:22:28: Cannot retrieve data from ADS due to rate limitations.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. AbstractIn 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.Popular summaryQuantum 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} }► References [1] Yujie Zhang, David Schmid, Yìlè Yīng, and Robert W. Spekkens. ``Reassessing the boundary between classical and nonclassical for individual quantum processes''. Phys. Rev. X 16, 021050 (2026). https://doi.org/10.1103/vqfz-wzjg [2] R. W. 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