Shadows and subsystems of generalized probabilistic theories: when tomographic incompleteness is not a loophole for contextuality proofs

David Schmid; John H. Selby; Vinicius P. Rossi; Roberto D. Baldijão; Ana Belén Sainz

doi:10.22331/q-2025-10-13-1880

Shadows and subsystems of generalized probabilistic theories: when tomographic incompleteness is not a loophole for contextuality proofs

David Schmid^1,2, John H. Selby¹, Vinicius P. Rossi¹, Roberto D. Baldijão^1,2, and Ana Belén Sainz^1,3

¹International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-309 Gdańsk, Poland
²Perimeter Institute for Theoretical Physics, N2L 2Y5 Waterloo, Canada
³Basic Research Community for Physics e.V., Germany

Published:	2025-10-13, volume 9, page 1880
Editor:	Cyril Branciard
Eprint:	arXiv:2409.13024v2
Doi:	https://doi.org/10.22331/q-2025-10-13-1880
Citation:	Quantum 9, 1880 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

It is commonly believed that failures of tomographic completeness undermine assessments of nonclassicality in noncontextuality experiments. In this work, we study how such failures can indeed lead to mistaken assessments of nonclassicality. We then show that proofs of the failure of noncontextuality are robust to a very broad class of failures of tomographic completeness, including the kinds of failures that are likely to occur in real experiments. We do so by showing that such proofs actually rely on a much weaker assumption that we term $\textit{relative tomographic completeness}$: namely, that one's experimental procedures are tomographic $\textit{for each other}$. Thus, the failure of noncontextuality can be established even with coarse-grained, effective, emergent, or virtual degrees of freedom. This also implies that the existence of a deeper theory of nature (beyond that being probed in one's experiment) does not in and of itself pose any challenge to proofs of nonclassicality. To prove these results, we first introduce a number of useful new concepts within the framework of generalized probabilistic theories (GPTs). Most notably, we introduce the notion of a GPT $subsystem$, generalizing a range of preexisting notions of subsystems (including those arising from tensor products, direct sums, decoherence processes, virtual encodings, and more). We also introduce the notion of a $shadow$ of a GPT fragment, which captures the information lost when one's states and effects are unknowingly not tomographic for one another.

► BibTeX data

@article{Schmid2025shadowssubsystemsof,
  doi = {10.22331/q-2025-10-13-1880},
  url = {https://doi.org/10.22331/q-2025-10-13-1880},
  title = {Shadows and subsystems of generalized probabilistic theories: when tomographic incompleteness is not a loophole for contextuality proofs},
  author = {Schmid, David and Selby, John H. and Rossi, Vinicius P. and Baldij{\~{a}}o, Roberto D. and Sainz, Ana Bel{\'{e}}n},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1880},
  month = oct,
  year = {2025}
}

► References

[1] R. W. Spekkens. ``Contextuality for preparations, transformations, and unsharp measurements''. Phys. Rev. A 71, 052108 (2005).
https://doi.org/10.1103/PhysRevA.71.052108

[2] Robert W. Spekkens. ``The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of Einstein'' (2019). arXiv:1909.04628.
arXiv:1909.04628

[3] David Schmid. ``Guiding our interpretation of quantum theory by principles of causation and inference''. PhD thesis, University of Waterloo (2021).

[4] Robert W. Spekkens. ``Negativity and Contextuality are Equivalent Notions of Nonclassicality''. Phys. Rev. Lett. 101, 020401 (2008).
https://doi.org/10.1103/PhysRevLett.101.020401

[5] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens. ``A structure theorem for generalized-noncontextual ontological models''. Quantum 8, 1283 (2024).
https://doi.org/10.22331/q-2024-03-14-1283

[6] David Schmid, John H. Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens. ``Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories''. PRX Quantum 2, 010331 (2021).
https://doi.org/10.1103/PRXQuantum.2.010331

[7] David Schmid, Haoxing Du, John H. Selby, and Matthew F. Pusey. ``Uniqueness of noncontextual models for stabilizer subtheories''. Phys. Rev. Lett. 129, 120403 (2022).
https://doi.org/10.1103/PhysRevLett.129.120403

[8] Farid Shahandeh. ``Quantum computational advantage implies contextuality'' (2021). arXiv:2112.00024.
arXiv:2112.00024

[9] David Schmid and Robert W Spekkens. ``Contextual advantage for state discrimination''. Phys. Rev. X 8, 011015 (2018).
https://doi.org/10.1103/PhysRevX.8.011015

[10] Kieran Flatt, Hanwool Lee, Carles Roch I Carceller, Jonatan Bohr Brask, and Joonwoo Bae. ``Contextual advantages and certification for maximum-confidence discrimination''. PRX Quantum 3, 030337 (2022).
https://doi.org/10.1103/PRXQuantum.3.030337

[11] Sumit Mukherjee, Shivam Naonit, and A. K. Pan. ``Discriminating three mirror-symmetric states with a restricted contextual advantage''. Phys. Rev. A 106, 012216 (2022).
https://doi.org/10.1103/PhysRevA.106.012216

[12] Jaehee Shin, Donghoon Ha, and Younghun Kwon. ``Quantum contextual advantage depending on nonzero prior probabilities in state discrimination of mixed qubit states''. Entropy 23, 1583 (2021).
https://doi.org/10.3390/e23121583

[13] Lorenzo Catani, Matthew Leifer, David Schmid, and Robert W. Spekkens. ``Why interference phenomena do not capture the essence of quantum theory''. Quantum 7, 1119 (2023).
https://doi.org/10.22331/q-2023-09-25-1119

[14] Lorenzo Catani, Matthew Leifer, David Schmid, and Robert W Spekkens. ``Reply to ``Comment on `Why interference phenomena do not capture the essence of quantum theory''''' (2022). arXiv:2207.11791.
arXiv:2207.11791

[15] Lorenzo Catani, Matthew Leifer, Giovanni Scala, David Schmid, and Robert W. Spekkens. ``Aspects of the phenomenology of interference that are genuinely nonclassical''. Phys. Rev. A 108, 022207 (2023).
https://doi.org/10.1103/PhysRevA.108.022207

[16] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. ``Contextuality without incompatibility''. Phys. Rev. Lett. 130, 230201 (2023).
https://doi.org/10.1103/PhysRevLett.130.230201

[17] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. ``Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality''. Phys. Rev. A 107, 062203 (2023).
https://doi.org/10.1103/PhysRevA.107.062203

[18] Armin Tavakoli and Roope Uola. ``Measurement incompatibility and steering are necessary and sufficient for operational contextuality''. Phys. Rev. Research 2, 013011 (2020).
https://doi.org/10.1103/PhysRevResearch.2.013011

[19] Lorenzo Catani, Matthew Leifer, Giovanni Scala, David Schmid, and Robert W. Spekkens. ``What is nonclassical about uncertainty relations?''. Phys. Rev. Lett. 129, 240401 (2022).
https://doi.org/10.1103/PhysRevLett.129.240401

[20] Matteo Lostaglio. ``Certifying Quantum Signatures in Thermodynamics and Metrology via Contextuality of Quantum Linear Response''. Phys. Rev. Lett. 125, 230603 (2020).
https://doi.org/10.1103/PhysRevLett.125.230603

[21] Naim Elias Comar, Danilo Cius, Luis Felipe Santos, Rafael Wagner, and Bárbara Amaral. ``Contextuality in anomalous heat flow'' (2024). arXiv:2406.09715.
arXiv:2406.09715

[22] Matthew F. Pusey. ``Anomalous Weak Values Are Proofs of Contextuality''. Phys. Rev. Lett. 113, 200401 (2014).
https://doi.org/10.1103/PhysRevLett.113.200401

[23] Ravi Kunjwal, Matteo Lostaglio, and Matthew F. Pusey. ``Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts''. Phys. Rev. A 100, 042116 (2019).
https://doi.org/10.1103/PhysRevA.100.042116

[24] Vinicius P. Rossi, David Schmid, John H. Selby, and Ana Belén Sainz. ``Contextuality with vanishing coherence and maximal robustness to dephasing''. Phys. Rev. A 108, 032213 (2023).
https://doi.org/10.1103/PhysRevA.108.032213

[25] Rafael Wagner, Anita Camillini, and Ernesto F. Galvão. ``Coherence and contextuality in a Mach-Zehnder interferometer''. Quantum 8, 1240 (2024).
https://doi.org/10.22331/q-2024-02-05-1240

[26] Roberto D. Baldijão, Rafael Wagner, Cristhiano Duarte, Bárbara Amaral, and Marcelo Terra Cunha. ``Emergence of Noncontextuality under Quantum Darwinism''. PRX Quantum 2, 030351 (2021).
https://doi.org/10.1103/PRXQuantum.2.030351

[27] Robert W. Spekkens, D. H. Buzacott, A. J. Keehn, Ben Toner, and G. J. Pryde. ``Preparation Contextuality Powers Parity-Oblivious Multiplexing''. Phys. Rev. Lett. 102, 010401 (2009).
https://doi.org/10.1103/PhysRevLett.102.010401

[28] Andre Chailloux, Iordanis Kerenidis, Srijita Kundu, and Jamie Sikora. ``Optimal bounds for parity-oblivious random access codes''. New J. Phys. 18, 045003 (2016).
https://doi.org/10.1088/1367-2630/18/4/045003

[29] Andris Ambainis, Manik Banik, Anubhav Chaturvedi, Dmitry Kravchenko, and Ashutosh Rai. ``Parity oblivious d-level random access codes and class of noncontextuality inequalities''. Quantum Inf. Process. 18, 111 (2019).
https://doi.org/10.1007/s11128-019-2228-3

[30] Debashis Saha, Paweł Horodecki, and Marcin Pawłowski. ``State independent contextuality advances one-way communication''. New J. Phys. 21, 093057 (2019).
https://doi.org/10.1088/1367-2630/ab4149

[31] Shiv Akshar Yadavalli and Ravi Kunjwal. ``Contextuality in entanglement-assisted one-shot classical communication''. Quantum 6, 839 (2022).
https://doi.org/10.22331/q-2022-10-13-839

[32] Alley Hameedi, Armin Tavakoli, Breno Marques, and Mohamed Bourennane. ``Communication games reveal preparation contextuality''. Phys. Rev. Lett. 119, 220402 (2017).
https://doi.org/10.1103/PhysRevLett.119.220402

[33] Amanda M. Fonseca, Vinicius P. Rossi, Roberto D. Baldijão, John H. Selby, and Ana Belén Sainz. ``Robustness of contextuality under different types of noise as quantifiers for parity-oblivious multiplexing tasks''. Phys. Rev. A 111, 022217 (2025).
https://doi.org/10.1103/PhysRevA.111.022217

[34] Matteo Lostaglio and Gabriel Senno. ``Contextual advantage for state-dependent cloning''. Quantum 4, 258 (2020).
https://doi.org/10.22331/q-2020-04-27-258

[35] Pauli Jokinen, Mirjam Weilenmann, Martin Plávala, Juha-Pekka Pellonpää, Jukka Kiukas, and Roope Uola. ``No-broadcasting characterizes operational contextuality'' (2024). arXiv:2406.07305.
arXiv:2406.07305

[36] Victoria J. Wright and Máté Farkas. ``Invertible map between bell nonlocal and contextuality scenarios''. Phys. Rev. Lett. 131, 220202 (2023).
https://doi.org/10.1103/PhysRevLett.131.220202

[37] David Schmid, John H. Selby, and Robert W. Spekkens. ``Unscrambling the omelette of causation and inference: The framework of causal-inferential theories'' (2021). arXiv:2009.03297.
arXiv:2009.03297

[38] Ravi Kunjwal and Robert W. Spekkens. ``From the Kochen-Specker Theorem to Noncontextuality Inequalities without Assuming Determinism''. Phys. Rev. Lett. 115, 110403 (2015).
https://doi.org/10.1103/PhysRevLett.115.110403

[39] Ravi Kunjwal and Robert W. Spekkens. ``From statistical proofs of the kochen-specker theorem to noise-robust noncontextuality inequalities''. Phys. Rev. A 97, 052110 (2018).
https://doi.org/10.1103/PhysRevA.97.052110

[40] Ravi Kunjwal. ``Contextuality beyond the Kochen-Specker theorem'' (2016). arXiv:1612.07250.
arXiv:1612.07250

[41] Ravi Kunjwal. ``Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements''. Quantum 3, 184 (2019).
https://doi.org/10.22331/q-2019-09-09-184

[42] Ravi Kunjwal. ``Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem''. Quantum 4, 219 (2020).
https://doi.org/10.22331/q-2020-01-10-219

[43] Ravi Kunjwal. ``Fine's theorem, noncontextuality, and correlations in specker's scenario''. Phys. Rev. A 91, 022108 (2015).
https://doi.org/10.1103/PhysRevA.91.022108

[44] Tomáš Gonda, Ravi Kunjwal, David Schmid, Elie Wolfe, and Ana Belén Sainz. ``Almost Quantum Correlations are Inconsistent with Specker's Principle''. Quantum 2, 87 (2018).
https://doi.org/10.22331/q-2018-08-27-87

[45] Michael D. Mazurek, Matthew F. Pusey, Kevin J. Resch, and Robert W. Spekkens. ``Experimentally Bounding Deviations From Quantum Theory in the Landscape of Generalized Probabilistic Theories''. PRX Quantum 2, 020302 (2021).
https://doi.org/10.1103/PRXQuantum.2.020302

[46] David Schmid, John H. Selby, and Robert W. Spekkens. ``Addressing some common objections to generalized noncontextuality''. Phys. Rev. A 109, 022228 (2024).
https://doi.org/10.1103/PhysRevA.109.022228

[47] Markus P. Müller and Andrew J. P. Garner. ``Testing quantum theory by generalizing noncontextuality''. Phys. Rev. X 13, 041001 (2023).
https://doi.org/10.1103/PhysRevX.13.041001

[48] L. Hardy. ``Quantum Theory From Five Reasonable Axioms'' (2001). arXiv:quant-ph/0101012.
arXiv:quant-ph/0101012

[49] Jonathan Barrett. ``Information processing in generalized probabilistic theories''. Phys. Rev. A 75, 032304 (2007).
https://doi.org/10.1103/PhysRevA.75.032304

[50] Peter Janotta and Raymond Lal. ``Generalized probabilistic theories without the no-restriction hypothesis''. Phys. Rev. A 87, 052131 (2013).
https://doi.org/10.1103/PhysRevA.87.052131

[51] Andrew J. P. Garner and Markus P. Mueller. ``Characterization of the probabilistic models that can be embedded in quantum theory'' (2020). arXiv:2004.06136.
arXiv:2004.06136

[52] Michael D Mazurek, Matthew F Pusey, Ravi Kunjwal, Kevin J Resch, and Robert W Spekkens. ``An experimental test of noncontextuality without unphysical idealizations''. Nature communications 7, 1–7 (2016).
https://doi.org/10.1038/ncomms11780

[53] Matthew F. Pusey, Lídia del Rio, and Bettina Meyer. ``Contextuality without access to a tomographically complete set'' (2019). arXiv:1904.08699.
arXiv:1904.08699

[54] David Schmid, Roberto D Baldijão, John H Selby, Ana Belén Sainz, and Robert W Spekkens. ``Noncontextuality inequalities for prepare-transform-measure scenarios'' (2024). arXiv:2407.09624.
arXiv:2407.09624

[55] Lluís Masanes and Markus P Müller. ``A derivation of quantum theory from physical requirements''. New Journal of Physics 13, 063001 (2011).
https://doi.org/10.1088/1367-2630/13/6/063001

[56] Martin Plávala. ``General probabilistic theories: An introduction'' (2021). arXiv:2103.07469.
https://doi.org/10.1016/j.physrep.2023.09.001
arXiv:2103.07469

[57] 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.132.050202

[58] A. J. Leggett and Anupam Garg. ``Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?''. Phys. Rev. Lett. 54, 857–860 (1985).
https://doi.org/10.1103/PhysRevLett.54.857

[59] David Schmid. ``A review and reformulation of macroscopic realism: resolving its deficiencies using the framework of generalized probabilistic theories''. Quantum 8, 1217 (2024).
https://doi.org/10.22331/q-2024-01-03-1217

[60] Daniel Gottesman. ``Stabilizer codes and quantum error correction'' (1997). arXiv:quant-ph/9705052.
arXiv:quant-ph/9705052

[61] Daniel Gottesman. ``The Heisenberg representation of quantum computers'' (1998). arXiv:quant-ph/9807006.
arXiv:quant-ph/9807006

[62] Ernst CG Stueckelberg. ``Quantum theory in real hilbert space''. Helv. Phys. Acta 33, 458 (1960).
https://doi.org/10.5169/seals-113093

[63] Carlton M Caves, Christopher A Fuchs, and Pranaw Rungta. ``Entanglement of formation of an arbitrary state of two rebits''. Foundations of Physics Letters 14, 199–212 (2001).
https://doi.org/10.1023/A:1012215309321

[64] Howard Barnum and Alexander Wilce. ``Information Processing in Convex Operational Theories''. Elec. Notes Theo. Comp. Sci. 270, 3–15 (2011).
https://doi.org/10.1016/j.entcs.2011.01.002

[65] Stephen D Bartlett, Terry Rudolph, and Robert W Spekkens. ``Reference frames, superselection rules, and quantum information''. Reviews of Modern Physics 79, 555 (2007).
https://doi.org/10.1103/RevModPhys.79.555

[66] Paolo Zanardi. ``Virtual quantum subsystems''. Physical Review Letters 87, 077901 (2001).
https://doi.org/10.1103/PhysRevLett.87.077901

[67] Benjamin Schumacher. ``Quantum coding''. Phys. Rev. A 51, 2738–2747 (1995).
https://doi.org/10.1103/PhysRevA.51.2738

[68] Daniel A. Lidar and K. Birgitta Whaley. ``Decoherence-free subspaces and subsystems''. Pages 83–120. Springer Berlin Heidelberg. Berlin, Heidelberg (2003).
https://doi.org/10.1007/3-540-44874-8_5

[69] Ognyan Oreshkov. ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics''. Quantum 3, 206 (2019).
https://doi.org/10.22331/q-2019-12-02-206

[70] Karol Życzkowski. ``Quartic quantum theory: an extension of the standard quantum mechanics''. Journal of Physics A: Mathematical and Theoretical 41, 355302 (2008).
https://doi.org/10.1088/1751-8113/41/35/355302

[71] Jonathan G. Richens, John H. Selby, and Sabri W. Al-Safi. ``Entanglement is necessary for emergent classicality in all physical theories''. Phys. Rev. Lett. 119, 080503 (2017).
https://doi.org/10.1103/PhysRevLett.119.080503

[72] Ciarán M Lee and John H Selby. ``A no-go theorem for theories that decohere to quantum mechanics''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, 20170732 (2018).
https://doi.org/10.1098/rspa.2017.0732

[73] James Hefford and Stefano Gogioso. ``Hyper-decoherence in density hypercubes''. Electronic Proceedings in Theoretical Computer Science 340, 141–159 (2021).
https://doi.org/10.4204/eptcs.340.7

[74] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Probabilistic theories with purification''. Phys. Rev. A 81, 062348 (2010).
https://doi.org/10.1103/PhysRevA.81.062348

[75] Theodore W. Gamelin and Robert Everist Greene. ``Introduction to topology''. Dover Publications. Mineola, N.Y. (1999, 2nd ed.).
https://store.doverpublications.com/products/9780486320182?_pos=1&_sid=f1329e25e&_ss=r

[76] Victor Gitton and Mischa P. Woods. ``Solvable criterion for the contextuality of any prepare-and-measure scenario''. Quantum 6, 732 (2022).
https://doi.org/10.22331/q-2022-06-07-732

[77] Alexander Holevo. ``Probabilistic and statistical aspects of quantum theory''. Edizioni della Normale. (2011).
https://doi.org/10.1007/978-88-7642-378-9

[78] E G Beltrametti and S Bugajski. ``A classical extension of quantum mechanics''. J. Phys. A 28, 3329–3343 (1995).
https://doi.org/10.1088/0305-4470/28/12/007

[79] Michael J. Grabowecky, Christopher A. J. Pollack, Andrew R. Cameron, Robert W. Spekkens, and Kevin J. Resch. ``Experimentally bounding deviations from quantum theory for a photonic three-level system using theory-agnostic tomography''. Phys. Rev. A 105, 032204 (2022).
https://doi.org/10.1103/PhysRevA.105.032204

[80] Daniel Centeno, Marco Erba, David Schmid, John H. Selby, Robert W. Spekkens, Sina Soltani, Jacopo Surace, Alex Wilce, and Yìlè Yīng. ``Twirled worlds: symmetry-induced failures of tomographic locality'' (2024). arXiv:2407.21688.
arXiv:2407.21688

[81] Giulio Chiribella. ``Agents, subsystems, and the conservation of information''. Entropy 20, 358 (2018).
https://doi.org/10.3390/e20050358

[82] Lea Krämer and Lídia Del Rio. ``Operational locality in global theories''. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20170321 (2018).
https://doi.org/10.1098/rsta.2017.0321

[83] Victor Gitton and Mischa P. Woods. ``On the system loophole of generalized noncontextuality'' (2022). arXiv:2209.04469.
https://doi.org/10.1103/PhysRevResearch.6.043289
arXiv:2209.04469

[84] Simon Kochen and Ernst Specker. ``The problem of hidden variables in quantum mechanics''. J. Math. & Mech. 17, 59–87 (1967).
https://doi.org/10.1007/978-94-010-1795-4_17

[85] Charalambos D Aliprantis. ``Principles of real analysis''. Academic Press. San Diego, CA (1998). 3 edition.
https://shop.elsevier.com/books/principles-of-real-analysis/burkinshaw/978-0-08-057310-6

[86] Robert Lang. ``A note on the measurability of convex sets''. Archiv der Mathematik 47, 90–92 (1986).
https://doi.org/10.1007/BF01202504

[87] Keith Conrad. ``Lecture Notes on Bilinear Forms''.
https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf

Cited by

[1] Sina Soltani, Marco Erba, David Schmid, and John H.Selby, "Decoupling local classicality from classical explainability: A noncontextual model for bilocal classical theory and a locally-classical but contextual theory", Quantum 10, 2142 (2026).

[2] 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).

[3] Yujie Zhang, Yìlè Yīng, and David Schmid, "Quantifiers and witnesses for the nonclassicality of measurements and of states", Quantum 10, 2180 (2026).

[4] Philip A. LeMaitre, "Harvesting contextuality from the vacuum", Physical Review D 113 4, 045001 (2026).

[5] Albert Aloy, Matteo Fadel, Thomas D. Galley, Caroline L. Jones, and Markus P. Müller, "Theory-independent monitoring of the decoherence of a superconducting qubit with generalized contextuality", Nature Communications 17 1, 2474 (2026).

[6] Davide Rolino, Marco Erba, Alessandro Tosini, and Paolo Perinotti, "Minimal operational theories: classical theories with quantum features", New Journal of Physics 27 2, 023004 (2025).

[7] 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).

[8] Rafael Wagner, "Coherence and contextuality as quantum resources", arXiv:2511.16785, (2025).

[9] Mina Doosti, Theodoros Yianni, and Farid Shahandeh, "Contextuality of all optimal quantum cloning", arXiv:2406.19382, (2024).

[10] Nick Ormrod, "A causal derivation of the algebraic approach to quantum systems", arXiv:2508.01111, (2025).

[11] Farid Shahandeh, Theodoros Yianni, and Mina Doosti, "A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality", arXiv:2512.10000, (2025).

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

[13] Enrico Bozzetto and Jonte R. Hance, "Warring Contextualities -- Provably Classical vs Provably Nonclassical", arXiv:2604.14319, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 07:55:44) and SAO/NASA ADS (last updated successfully 2026-08-08 19:00:24). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-09 07:55:44: 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.