Self-Testing Graph States Permitting Bounded Classical Communication
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AbstractSelf-testing identifies quantum states and correlations that exhibit nonlocality, distinguishing them, up to local transformations, from other quantum states. Due to their strong nonlocality, it is known that all graph states can be self-tested in the standard setting – where parties are not allowed to communicate. Recently it has been shown that graph states display nonlocal correlations even when bounded classical communication on the underlying graph is permitted, a feature that has found applications in proving a circuit-depth separation between classical and quantum computing. In this work, we develop self testing in the framework of bounded classical communication, and we show that certain graph states can be robustly self-tested even allowing for communication. In particular, we provide an explicit self-test for the circular graph state and the honeycomb cluster state – the latter known to be a universal resource for measurement based quantum computation. Since communication generally obstructs self-testing of graph states, we further provide a procedure to robustly self-test any graph state from larger ones that exhibit nonlocal correlations in the communication scenario.► BibTeX data@article{Meyer2026selftestinggraph, doi = {10.22331/q-2026-01-08-1961}, url = {https://doi.org/10.22331/q-2026-01-08-1961}, title = {Self-{T}esting {G}raph {S}tates {P}ermitting {B}ounded {C}lassical {C}ommunication}, author = {Meyer, Uta Isabella and {\v{S}}upi{\'{c}}, Ivan and Grosshans, Fr{\'{e}}d{\'{e}}ric and Markham, Damian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1961}, month = jan, year = {2026} }► References [1] John S Bell. ``On the Einstein Podolsky Rosen paradox''.
Physics Physique Fizika 1, 195–200 (1964). https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 [2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Rev. Mod. Phys. 86, 419–478 (2014). https://doi.org/10.1103/RevModPhys.86.419 [3] Roger Colbeck. ``Quantum and relativistic protocols for secure multi-party computation''. PhD thesis. University of Cambridge. (2006). url: https://arxiv.org/abs/0911.3814. arXiv:0911.3814 [4] Roger Colbeck and Adrian Kent. ``Private randomness expansion with untrusted devices''. Journal of Physics A: Mathematical and Theoretical 44, 095305 (2011). https://doi.org/10.1088/1751-8113/44/9/095305 [5] Antonio Acín and Lluis Masanes. ``Certified randomness in quantum physics''. 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Could not fetch ADS cited-by data during last attempt 2026-01-08 14:33:41: 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. AbstractSelf-testing identifies quantum states and correlations that exhibit nonlocality, distinguishing them, up to local transformations, from other quantum states. Due to their strong nonlocality, it is known that all graph states can be self-tested in the standard setting – where parties are not allowed to communicate. Recently it has been shown that graph states display nonlocal correlations even when bounded classical communication on the underlying graph is permitted, a feature that has found applications in proving a circuit-depth separation between classical and quantum computing. In this work, we develop self testing in the framework of bounded classical communication, and we show that certain graph states can be robustly self-tested even allowing for communication. In particular, we provide an explicit self-test for the circular graph state and the honeycomb cluster state – the latter known to be a universal resource for measurement based quantum computation. Since communication generally obstructs self-testing of graph states, we further provide a procedure to robustly self-test any graph state from larger ones that exhibit nonlocal correlations in the communication scenario.► BibTeX data@article{Meyer2026selftestinggraph, doi = {10.22331/q-2026-01-08-1961}, url = {https://doi.org/10.22331/q-2026-01-08-1961}, title = {Self-{T}esting {G}raph {S}tates {P}ermitting {B}ounded {C}lassical {C}ommunication}, author = {Meyer, Uta Isabella and {\v{S}}upi{\'{c}}, Ivan and Grosshans, Fr{\'{e}}d{\'{e}}ric and Markham, Damian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1961}, month = jan, year = {2026} }► References [1] John S Bell. ``On the Einstein Podolsky Rosen paradox''.
Physics Physique Fizika 1, 195–200 (1964). https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 [2] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Rev. Mod. Phys. 86, 419–478 (2014). https://doi.org/10.1103/RevModPhys.86.419 [3] Roger Colbeck. ``Quantum and relativistic protocols for secure multi-party computation''. PhD thesis. University of Cambridge. (2006). url: https://arxiv.org/abs/0911.3814. arXiv:0911.3814 [4] Roger Colbeck and Adrian Kent. ``Private randomness expansion with untrusted devices''. Journal of Physics A: Mathematical and Theoretical 44, 095305 (2011). https://doi.org/10.1088/1751-8113/44/9/095305 [5] Antonio Acín and Lluis Masanes. ``Certified randomness in quantum physics''. Nature 540, 213–219 (2016). https://doi.org/10.1038/nature20119 [6] Stefano Pironio, Antonio Acín, Serge Massar, A Boyer de La Giroday, Dzimitry N Matsukevich, Peter Maunz, Steven Olmschenk, David Hayes, Le Luo, T Andrew Manning, et al. ``Random numbers certified by Bell's theorem''. Nature 464, 1021–1024 (2010). https://doi.org/10.1038/nature09008 [7] Antonio Acín, Nicolas Gisin, and Lluis Masanes. ``From bell's theorem to secure quantum key distribution''. Phys. Rev. Lett. 97, 120405 (2006). https://doi.org/10.1103/PhysRevLett.97.120405 [8] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. ``Device-independent security of quantum cryptography against collective attacks''. Phys. Rev. Lett. 98, 230501 (2007). https://doi.org/10.1103/PhysRevLett.98.230501 [9] Dominic Mayers and Andrew Yao. ``Quantum cryptography with imperfect apparatus''. In Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280). Pages 503–509. (1998). https://doi.org/10.1109/SFCS.1998.743501 [10] Dominic Mayers and Andrew Yao. ``Self testing quantum apparatus''. Quantum Information & Computation 4, 273–286 (2004). https://doi.org/10.26421/QIC4.4-3 [11] Ivan Šupić and Joseph Bowles. ``Self-testing of quantum systems: a review''. Quantum 4, 337 (2020). https://doi.org/10.22331/q-2020-09-30-337 [12] Umesh Vazirani and Thomas Vidick. ``Fully device independent quantum key distribution''. Commun. ACM 62, 133 (2019). https://doi.org/10.1145/3310974 [13] Rotem Arnon-Friedman, Frédéric Dupuis, Omar Fawzi, Renato Renner, and Thomas Vidick. ``Practical device-independent quantum cryptography via entropy accumulation''. Nature communications 9, 459 (2018). https://doi.org/10.1038/s41467-017-02307-4 [14] Rodrigo Gallego, Nicolas Brunner, Christopher Hadley, and Antonio Acín. ``Device-independent tests of classical and quantum dimensions''. Phys. Rev. Lett. 105, 230501 (2010). https://doi.org/10.1103/PhysRevLett.105.230501 [15] Andrea Coladangelo, Alex B Grilo, Stacey Jeffery, and Thomas Vidick. ``Verifier-on-a-leash: new schemes for verifiable delegated quantum computation, with quasilinear resources''. In Annual international conference on the theory and applications of cryptographic techniques. Pages 247–277. Springer (2019). https://doi.org/10.1007/978-3-030-17659-4_9 [16] Ben W Reichardt, Falk Unger, and Umesh Vazirani. ``Classical command of quantum systems''. Nature 496, 456–460 (2013). https://doi.org/10.1038/nature12035 [17] Jérémy Ribeiro, Gláucia Murta, and Stephanie Wehner. ``Fully device-independent conference key agreement''. Physical Review A 97, 022307 (2018). https://doi.org/10.1103/PhysRevA.97.022307 [18] Kim Vallée, Pierre-Emmanuel Emeriau, Boris Bourdoncle, Adel Sohbi, Shane Mansfield, and Damian Markham. ``Corrected bell and non-contextuality inequalities for realistic experiments''. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 382, 20230011 (2024). https://doi.org/10.1098/rsta.2023.0011 [19] Andreas Fyrillas, Boris Bourdoncle, Alexandre Maïnos, Pierre-Emmanuel Emeriau, Kayleigh Start, Nico Margaria, Martina Morassi, Aristide Lemaı̂tre, Isabelle Sagnes, Petr Stepanov, Thi Huong Au, Sébastien Boissier, Niccolo Somaschi, Nicolas Maring, Nadia Belabas, and Shane Mansfield. ``Certified randomness in tight space''. PRX Quantum 5, 020348 (2024). https://doi.org/10.1103/PRXQuantum.5.020348 [20] Gláucia Murta and Flavio Baccari. ``Self-testing with dishonest parties and device-independent entanglement certification in quantum communication networks''. Phys. Rev. Lett. 131, 140201 (2023). https://doi.org/10.1103/PhysRevLett.131.140201 [21] Richard Cleve, Peter Hoyer, Benjamin Toner, and John Watrous. ``Consequences and limits of nonlocal strategies''. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004. Pages 236–249. IEEE (2004). https://doi.org/10.1109/CCC.2004.1313847 [22] Robert Raussendorf and Hans J. Briegel. ``A one-way quantum computer''. Phys. Rev. Lett. 86, 5188–5191 (2001). https://doi.org/10.1103/PhysRevLett.86.5188 [23] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al. ``Fusion-based quantum computation''. Nature Communications 14, 912 (2023). https://doi.org/10.1038/s41467-023-36493-1 [24] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. Ph. D. thesis, California Institute of Technology (1997). https://doi.org/10.48550/arXiv.quant-ph/9705052 arXiv:quant-ph/9705052 [25] Robert Raussendorf, Jim Harrington, and Kovid Goyal. ``A fault-tolerant one-way quantum computer''. Annals of physics 321, 2242–2270 (2006). https://doi.org/10.1016/j.aop.2006.01.012 [26] Nathan Shettell and Damian Markham. ``Graph states as a resource for quantum metrology''. Phys. Rev. Lett. 124, 110502 (2020). https://doi.org/10.1103/PhysRevLett.124.110502 [27] Damian Markham and Barry C. Sanders. ``Graph states for quantum secret sharing''. Phys. Rev. A 78, 042309 (2008). https://doi.org/10.1103/PhysRevA.78.042309 [28] Matthias Christandl and Stephanie Wehner. ``Quantum anonymous transmissions''. In International conference on the theory and application of cryptology and information security. Pages 217–235. Springer (2005). https://doi.org/10.1007/11593447_12 [29] Chao-Yang Lu, Xiao-Qi Zhou, Otfried Gühne, Wei-Bo Gao, Jin Zhang, Zhen-Sheng Yuan, Alexander Goebel, Tao Yang, and Jian-Wei Pan. ``Experimental entanglement of six photons in graph states''. Nature physics 3, 91–95 (2007). https://doi.org/10.1038/nphys507 [30] Ido Schwartz, Dan Cogan, Emma R Schmidgall, Yaroslav Don, Liron Gantz, Oded Kenneth, Netanel H Lindner, and David Gershoni. ``Deterministic generation of a cluster state of entangled photons''. Science 354, 434–437 (2016). https://doi.org/10.1126/science.aah4758 [31] D Istrati, Y Pilnyak, JC Loredo, C Antón, N Somaschi, P Hilaire, H Ollivier, M Esmann, L Cohen, L Vidro, et al. ``Sequential generation of linear cluster states from a single photon emitter''. Nature communications 11, 5501 (2020). https://doi.org/10.1038/s41467-020-19341-4 [32] Bryn A Bell, DA Herrera-Martí, MS Tame, Damian Markham, WJ Wadsworth, and JG Rarity. ``Experimental demonstration of a graph state quantum error-correction code''. 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