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Resource-theoretic hierarchy of contextuality for general probabilistic theories

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Resource-theoretic hierarchy of contextuality for general probabilistic theories

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AbstractIn this work we present a hierarchy of generalized contextuality. It refines the traditional binary distinction between contextual and noncontextual theories, and facilitates their comparison based on how contextual they are. Our approach focuses on the contextuality of prepare-and-measure scenarios, described by general probabilistic theories (GPTs). To motivate the hierarchy, we define it as the resource ordering of a novel resource theory of GPT-contextuality. The building blocks of its free operations are classical systems and univalent simulations between GPTs. These simulations preserve operational equivalences and thus cannot generate contextuality. Noncontextual theories can be recovered as least elements in the hierarchy. We then define a new contextuality monotone, called classical excess, given by the minimal error of embedding a GPT within an infinite classical system. In addition, we show that the optimal success probability in the parity oblivious multiplexing game also defines a monotone in our resource theory. Finally, we discuss whether the non-free operations can be understood as implementing information erasure and thus explaining the fine-tuning aspect of contextuality.Popular summaryA central question about quantum theory is which of its features truly constitute a departure from the classical worldview. A well-known and suitable criterion is that of generalized contextuality, which expresses the idea that procedures indistinguishable by experiment should not be treated as different in an underlying explanation. It applies broadly, is experimentally testable, and matters for practical tasks in quantum information processing. Rather than a simple yes/no test for contextuality, studied traditionally, this paper develops a precise way to address the above question by comparing physical theories according to how far they depart from the classical idea of noncontextuality. The hierarchy of physical systems defined here is motivated by a resource theory. The objects being compared are individual systems described within the formalism of general probabilistic theories, such as classical or quantum systems of a given dimension. The free transformations include simulations that preserve the indistinguishability relations defining noncontextuality and also allow access to all classical systems. This yields a principled way to compare contextual theories, while ensuring that all noncontextual systems lie at the bottom of the hierarchy. The authors further study the resulting contextuality monotones—numerical measures that cannot increase under the free operations—and in particular define the “classical excess,” which captures the smallest error with which a given theory can be simulated by a classical system. They also show that the optimal success probability in a standard communication task, parity‑oblivious multiplexing, yields another monotone that reflects contextual advantage in information‑processing tasks. Finally, the work explores a conceptual interpretation in which the non‑free simulations might be seen as processes that erase distinctions at a deeper ontological level so that they become operationally indistinguishable. This could help explain why certain ontological differences would not show up in experiments and suggest novel avenues to address the interpretational issues connected to the contextuality of quantum theory. Overall, the paper supplies a clear resource‑theoretic language and concrete measures for comparing how “nonclassical” different theories are, moving beyond the binary label “contextual vs noncontextual” and providing tools to quantify and compare contextuality in settings of interest for quantum information.► BibTeX data@article{Catani2026resourcetheoretic, doi = {10.22331/q-2026-04-21-2077}, url = {https://doi.org/10.22331/q-2026-04-21-2077}, title = {Resource-theoretic hierarchy of contextuality for general probabilistic theories}, author = {Catani, Lorenzo and Galley, Thomas D. and Gonda, Tom{\'{a}}{\v{s}}}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2077}, month = apr, year = {2026} }► References [1] David Schmid. Generalized noncontextuality. 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Quantum, 4: 345, October 2020. 10.22331/q-2020-10-21-345. https://doi.org/10.22331/q-2020-10-21-345Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-21 10:28:39: Could not fetch cited-by data for 10.22331/q-2026-04-21-2077 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-21 10:28:40: No response from ADS or unable to decode the received json data when getting the list of citing works.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 this work we present a hierarchy of generalized contextuality. It refines the traditional binary distinction between contextual and noncontextual theories, and facilitates their comparison based on how contextual they are. Our approach focuses on the contextuality of prepare-and-measure scenarios, described by general probabilistic theories (GPTs). To motivate the hierarchy, we define it as the resource ordering of a novel resource theory of GPT-contextuality. The building blocks of its free operations are classical systems and univalent simulations between GPTs. These simulations preserve operational equivalences and thus cannot generate contextuality. Noncontextual theories can be recovered as least elements in the hierarchy. We then define a new contextuality monotone, called classical excess, given by the minimal error of embedding a GPT within an infinite classical system. In addition, we show that the optimal success probability in the parity oblivious multiplexing game also defines a monotone in our resource theory. Finally, we discuss whether the non-free operations can be understood as implementing information erasure and thus explaining the fine-tuning aspect of contextuality.Popular summaryA central question about quantum theory is which of its features truly constitute a departure from the classical worldview. A well-known and suitable criterion is that of generalized contextuality, which expresses the idea that procedures indistinguishable by experiment should not be treated as different in an underlying explanation. It applies broadly, is experimentally testable, and matters for practical tasks in quantum information processing. Rather than a simple yes/no test for contextuality, studied traditionally, this paper develops a precise way to address the above question by comparing physical theories according to how far they depart from the classical idea of noncontextuality. The hierarchy of physical systems defined here is motivated by a resource theory. The objects being compared are individual systems described within the formalism of general probabilistic theories, such as classical or quantum systems of a given dimension. The free transformations include simulations that preserve the indistinguishability relations defining noncontextuality and also allow access to all classical systems. This yields a principled way to compare contextual theories, while ensuring that all noncontextual systems lie at the bottom of the hierarchy. The authors further study the resulting contextuality monotones—numerical measures that cannot increase under the free operations—and in particular define the “classical excess,” which captures the smallest error with which a given theory can be simulated by a classical system. They also show that the optimal success probability in a standard communication task, parity‑oblivious multiplexing, yields another monotone that reflects contextual advantage in information‑processing tasks. Finally, the work explores a conceptual interpretation in which the non‑free simulations might be seen as processes that erase distinctions at a deeper ontological level so that they become operationally indistinguishable. This could help explain why certain ontological differences would not show up in experiments and suggest novel avenues to address the interpretational issues connected to the contextuality of quantum theory. Overall, the paper supplies a clear resource‑theoretic language and concrete measures for comparing how “nonclassical” different theories are, moving beyond the binary label “contextual vs noncontextual” and providing tools to quantify and compare contextuality in settings of interest for quantum information.► BibTeX data@article{Catani2026resourcetheoretic, doi = {10.22331/q-2026-04-21-2077}, url = {https://doi.org/10.22331/q-2026-04-21-2077}, title = {Resource-theoretic hierarchy of contextuality for general probabilistic theories}, author = {Catani, Lorenzo and Galley, Thomas D. and Gonda, Tom{\'{a}}{\v{s}}}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2077}, month = apr, year = {2026} }► References [1] David Schmid. Generalized noncontextuality. Solstice of Foundations, ETH Zurich, 2022. https://www.youtube.com/watch?v=M3qn3EHWdOg. https://www.youtube.com/watch?v=M3qn3EHWdOg [2] R. W. Spekkens. Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A, 71: 052108, May 2005. 10.1103/PhysRevA.71.052108. https://doi.org/10.1103/PhysRevA.71.052108 [3] Robert Spekkens. The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of einstein. arXiv:1909.04628, 2019. https://doi.org/10.48550/arXiv.1909.04628. https://doi.org/10.48550/arXiv.1909.04628 arXiv:1909.04628 [4] Lorenzo Catani and Matthew Leifer. A mathematical framework for operational fine tunings. Quantum, 7: 948, March 2023. 10.22331/q-2023-03-16-948. https://doi.org/10.22331/q-2023-03-16-948 [5] Christopher Ferrie and Joseph Emerson. Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations. Journal of Physics A: Mathematical and Theoretical, 41 (35): 352001, jul 2008. 10.1088/1751-8113/41/35/352001. https://doi.org/10.1088/1751-8113/41/35/352001 [6] Robert W. Spekkens. Negativity and contextuality are equivalent notions of nonclassicality. Phys. Rev. Lett., 101: 020401, Jul 2008. 10.1103/PhysRevLett.101.020401. https://doi.org/10.1103/PhysRevLett.101.020401 [7] J. S. Bell. On the Einstein Podolsky Rosen paradox.

Closing Bell Boxing Black Box Simulations in the Resource Theory of Contextuality, pages 475–529.

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