Theory-independent randomness generation from spatial symmetries
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AbstractWe demonstrate a fundamental relation between the structures of physical space and of quantum theory: the set of quantum correlations in a rotational prepare-and-measure scenario can be derived from covariance alone, without assuming quantum physics. To show this, we consider a semi-device-independent randomness generation scheme where one of two spatial rotations is performed on an otherwise uncharacterized preparation device, and one of two possible measurement outcomes is subsequently obtained. An upper bound on a theory-independent notion of spin is assumed for the transmitted physical system. It turns out that this determines the set of quantum correlations and the amount of certifiable randomness in this setup exactly. Interestingly, this yields the basis of a theory-independent protocol for the secure generation of random numbers. Our results support the conjecture that the symmetries of space and time determine at least part of the probabilistic structure of quantum theory.Featured image: Our prepare-and-measure scenario, where the input corresponds to a spatial rotation of the preparation device.The following is an appropriate online talk for our paper, presented alongside follow-up work: International Symposium on Quantum Information and Communication (ISQIC), Kolkata, 31 March 2025 Popular summaryOur work explores the relationship between the symmetries of space and quantum probabilities. In particular, we explore how rotational symmetry around a fixed axis constrains the types of correlations one can expect to find in a simple “prepare-and-measure” experiment. We do so by bounding the spin of the communicated system, and compare the set of correlations as predicted by quantum theory to that of a more general, “post-quantum” set. Remarkably, we find that, for our 2-input 2-output scenario, the sets precisely coincide. That is to say, quantum physics is already the most general physical theory consistent with the symmetry of our scenario. Our work introduces “rotation boxes” as a direct analogue to the “non-local boxes” of the famous Bell experiment. This framework gives us a way to explore the most general statistical predictions given certain minimalist assumptions, motivated by spacetime. In the case of the Bell scenario, this is the no-signalling principle (that information cannot be communicated faster than the speed of light), from which it can be shown that quantum theory is a special case. In our case, we require only that the statistical response is covariant under rotations. Under this minimal symmetry assumption, we demonstrate that the set of quantum correlations is recovered precisely for our setup.► BibTeX data@article{Jones2026theoryindependent, doi = {10.22331/q-2026-01-16-1966}, url = {https://doi.org/10.22331/q-2026-01-16-1966}, title = {Theory-independent randomness generation from spatial symmetries}, author = {Jones, Caroline L. and Ludescher, Stefan L. and Aloy, Albert and M{\"{u}}ller, Markus P.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1966}, month = jan, year = {2026} }► References [1] L. Hardy, Quantum Theory From Five Reasonable Axioms, arXiv:quant-ph/0101012 (2001), DOI:10.48550/arXiv.quant-ph/0101012. https://doi.org/10.48550/arXiv.quant-ph/0101012 arXiv:quant-ph/0101012 [2] W. van Dam, Implausible consequences of superstrong nonlocality, Nat. Comput. 12, 9–12 (2013), DOI:10.1007/s11047-012-9353-6. https://doi.org/10.1007/s11047-012-9353-6 [3] M. Navascués and H. Wunderlich, A glance beyond the quantum model, Proc. R. Soc. Lond. A 466, 881–890 (2009), DOI:10.1098/rspa.2009.0453. https://doi.org/10.1098/rspa.2009.0453 [4] B. Dakić and Č. Brukner, Quantum theory and beyond: is entanglement special?, in Deep Beauty: Understanding the Quantum World through Mathematical Innovation (ed. H. Halvorson), Cambridge University Press, Cambridge (2011), DOI:10.1017/CBO9780511976971.011. https://doi.org/10.1017/CBO9780511976971.011 [5] G. Chiribella, G. M. d'Ariano, and P. Perinotti, Informational derivation of quantum theory, Phys. Rev. A 84, 012311 (2011), DOI:10.1103/PhysRevA.84.012311. https://doi.org/10.1103/PhysRevA.84.012311 [6] L. Masanes and M. P. Müller, A derivation of quantum theory from physical requirements, New J. 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Pironio, S. Massar, Security of practical private randomness generation, Phys. Rev. A 87(1), 012336 (2013), DOI:10.1103/PhysRevA.87.012336. https://doi.org/10.1103/PhysRevA.87.012336 [50] Y. Zhang, E. Knill, and P. Bierhorst, Certifying quantum randomness by probability estimation, Phys. Rev. A 98(4), 040304 (2018), DOI:10.1103/PhysRevA.98.040304. https://doi.org/10.1103/PhysRevA.98.040304 [51] E. Knill, Y. Zhang, P. Bierhorst, Generation of quantum randomness by probability estimation with classical side information, Phys. Rev. Research 2(3), 033465 (2020), DOI:10.1103/PhysRevResearch.2.033465. https://doi.org/10.1103/PhysRevResearch.2.033465Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-16 13:04:31: Could not fetch cited-by data for 10.22331/q-2026-01-16-1966 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-16 13:04:31: 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. AbstractWe demonstrate a fundamental relation between the structures of physical space and of quantum theory: the set of quantum correlations in a rotational prepare-and-measure scenario can be derived from covariance alone, without assuming quantum physics. To show this, we consider a semi-device-independent randomness generation scheme where one of two spatial rotations is performed on an otherwise uncharacterized preparation device, and one of two possible measurement outcomes is subsequently obtained. An upper bound on a theory-independent notion of spin is assumed for the transmitted physical system. It turns out that this determines the set of quantum correlations and the amount of certifiable randomness in this setup exactly. Interestingly, this yields the basis of a theory-independent protocol for the secure generation of random numbers. Our results support the conjecture that the symmetries of space and time determine at least part of the probabilistic structure of quantum theory.Featured image: Our prepare-and-measure scenario, where the input corresponds to a spatial rotation of the preparation device.The following is an appropriate online talk for our paper, presented alongside follow-up work: International Symposium on Quantum Information and Communication (ISQIC), Kolkata, 31 March 2025 Popular summaryOur work explores the relationship between the symmetries of space and quantum probabilities. In particular, we explore how rotational symmetry around a fixed axis constrains the types of correlations one can expect to find in a simple “prepare-and-measure” experiment. We do so by bounding the spin of the communicated system, and compare the set of correlations as predicted by quantum theory to that of a more general, “post-quantum” set. Remarkably, we find that, for our 2-input 2-output scenario, the sets precisely coincide. That is to say, quantum physics is already the most general physical theory consistent with the symmetry of our scenario. Our work introduces “rotation boxes” as a direct analogue to the “non-local boxes” of the famous Bell experiment. This framework gives us a way to explore the most general statistical predictions given certain minimalist assumptions, motivated by spacetime. In the case of the Bell scenario, this is the no-signalling principle (that information cannot be communicated faster than the speed of light), from which it can be shown that quantum theory is a special case. In our case, we require only that the statistical response is covariant under rotations. Under this minimal symmetry assumption, we demonstrate that the set of quantum correlations is recovered precisely for our setup.► BibTeX data@article{Jones2026theoryindependent, doi = {10.22331/q-2026-01-16-1966}, url = {https://doi.org/10.22331/q-2026-01-16-1966}, title = {Theory-independent randomness generation from spatial symmetries}, author = {Jones, Caroline L. and Ludescher, Stefan L. and Aloy, Albert and M{\"{u}}ller, Markus P.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1966}, month = jan, year = {2026} }► References [1] L. Hardy, Quantum Theory From Five Reasonable Axioms, arXiv:quant-ph/0101012 (2001), DOI:10.48550/arXiv.quant-ph/0101012. https://doi.org/10.48550/arXiv.quant-ph/0101012 arXiv:quant-ph/0101012 [2] W. van Dam, Implausible consequences of superstrong nonlocality, Nat. Comput. 12, 9–12 (2013), DOI:10.1007/s11047-012-9353-6. https://doi.org/10.1007/s11047-012-9353-6 [3] M. Navascués and H. Wunderlich, A glance beyond the quantum model, Proc. R. Soc. Lond. A 466, 881–890 (2009), DOI:10.1098/rspa.2009.0453. https://doi.org/10.1098/rspa.2009.0453 [4] B. Dakić and Č. Brukner, Quantum theory and beyond: is entanglement special?, in Deep Beauty: Understanding the Quantum World through Mathematical Innovation (ed. H. Halvorson), Cambridge University Press, Cambridge (2011), DOI:10.1017/CBO9780511976971.011. https://doi.org/10.1017/CBO9780511976971.011 [5] G. Chiribella, G. M. d'Ariano, and P. Perinotti, Informational derivation of quantum theory, Phys. Rev. A 84, 012311 (2011), DOI:10.1103/PhysRevA.84.012311. https://doi.org/10.1103/PhysRevA.84.012311 [6] L. Masanes and M. P. Müller, A derivation of quantum theory from physical requirements, New J. Phys. 13(6), 063001 (2011), DOI:10.1088/1367-2630/13/6/063001. https://doi.org/10.1088/1367-2630/13/6/063001 [7] S. Popescu, Nonlocality beyond quantum mechanics, Nat. Phys. 10, 264–270 (2014), DOI:10.1038/nphys2916. https://doi.org/10.1038/nphys2916 [8] M. P. Müller, Probabilistic Theories and Reconstructions of Quantum Theory, SciPost Phys. Lect. Notes 28 (2021), DOI:10.21468/SciPostPhysLectNotes.28. https://doi.org/10.21468/SciPostPhysLectNotes.28 [9] M. Plávala, General probabilistic theories: An introduction, Phys. Rep. 1033, 1–64 (2023), DOI:10.1016/j.physrep.2023.09.001. https://doi.org/10.1016/j.physrep.2023.09.001 [10] D. Mayers and A. Yao, Quantum cryptography with imperfect apparatus, Proceedings 39th Annual Symposium on Foundations of Computer Science (IEEE, 1998), 503–509, DOI:10.1109/SFCS.1998.743501. https://doi.org/10.1109/SFCS.1998.743501 [11] J. Barrett, L. Hardy, A. Kent, No signaling and quantum key distribution, Phys. Rev. Lett. 95, 010503 (2005), DOI:10.1103/PhysRevLett.95.010503. https://doi.org/10.1103/PhysRevLett.95.010503 [12] R. Colbeck, Quantum And Relativistic Protocols For Secure Multi-Party Computation, PhD thesis, University of Cambridge, 2006, DOI:10.48550/arXiv.0911.3814. https://doi.org/10.48550/arXiv.0911.3814 [13] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-independent security of quantum cryptography against collective attacks, Phys. Rev. Lett. 98, 230501 (2007), DOI:10.1103/PhysRevLett.98.230501. https://doi.org/10.1103/PhysRevLett.98.230501 [14] R. Gallego, N. Brunner, C. Hadley, and A. Acín, Device-independent tests of classical and quantum dimensions, Phys. Rev. Lett. 105, 230501 (2010), DOI:10.1103/PhysRevLett.105.230501. https://doi.org/10.1103/PhysRevLett.105.230501 [15] M. Pawłowski, and N. Brunner, Semi-device-independent security of one-way quantum key distribution, Phys. Rev. A 84, 010302 (2011), DOI:10.1103/PhysRevA.84.010302. https://doi.org/10.1103/PhysRevA.84.010302 [16] Y.-C. Liang, T. Vértesi, and N. Brunner, Semi-device-independent bounds on entanglement, Phys. Rev. A 83, 022108 (2011), DOI:10.1103/PhysRevA.83.022108. https://doi.org/10.1103/PhysRevA.83.022108 [17] C. Branciard, E. Cavalcanti, S. Walborn, V. Scarani, and H. M. Wiseman, One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering, Phys. Rev. A 85, 010301 (2012), DOI:10.1103/PhysRevA.85.010301. https://doi.org/10.1103/PhysRevA.85.010301 [18] T. Van Himbeeck, E. Woodhead, N. J. Cerf, R. García-Patrón, and S. Pironio, Semi-device-independent framework based on natural physical assumptions, Quantum 1, 33 (2017), DOI:10.22331/q-2017-11-18-33. https://doi.org/10.22331/q-2017-11-18-33 [19] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014), DOI:10.1103/RevModPhys.86.419. https://doi.org/10.1103/RevModPhys.86.419 [20] V. Scarani, Bell nonlocality, Oxford University Press, Oxford, 2019, DOI:10.1093/oso/9780198788416.001.0001. https://doi.org/10.1093/oso/9780198788416.001.0001 [21] H.-W. Li, Z.-Q. Yin, Y.-C. Wu, X.-B. Zou, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, Semi-device-independent random number expansion without entanglement, Phys. Rev. A 84, 034301 (2011), DOI:10.1103/PhysRevA.84.034301. https://doi.org/10.1103/PhysRevA.84.034301 [22] A. Acín and L. Masanes, Certified randomness in quantum physics, Nature 540, 213 (2016), DOI:10.1038/nature20119. https://doi.org/10.1038/nature20119 [23] X. Ma, X. Yuan, Z. Cao, B. Qi, and Z. Zhang Quantum random number generation, npj Quantum Inf. 2, 1 (2016), DOI:10.1038/npjqi.2016.21. https://doi.org/10.1038/npjqi.2016.21 [24] D. Rusca, T. van Himbeeck, A. Martin, J. B. Brask, W. Shi, S. Pironio, N. Brunner, and H. 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