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More entropy from shorter experiments using polytope approximations to the quantum set

Quantum Journal

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Researchers Hyejung H. Jee, Florian J. Curchod, and Mafalda L. Almeida developed a novel method to improve device-independent quantum random number generation (DI-QRNG) by using polytope approximations of the quantum set. Their approach employs two iterative algorithms that refine initial outer-polytope approximations, balancing computational efficiency with accuracy, guided by device behavior and cryptographic principles. Integrating these approximations with the probability estimation (PE) framework, the method achieves significantly higher certified entropy bounds in finite-size experiments, outperforming existing techniques with fewer device uses. Testing on bipartite and tripartite DI-QRNG protocols—using both simulated and experimental data—demonstrated notably faster entropy generation without added complexity, including for randomness amplification tasks. The team released their algorithms and PE tools under a non-commercial license, providing a ready-to-use solution for securing practical DI-QRNG protocols with improved entropy rates.

More entropy from shorter experiments using polytope approximations to the quantum set

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AbstractWe introduce a systematic method for constructing polytope approximations to the quantum set in a variety of device-independent quantum random number generation (DI-QRNG) protocols. Our approach relies on two general-purpose algorithms that iteratively refine an initial outer-polytope approximation, guided by typical device behaviour and cryptographic intuition. These refinements strike a balance between computational tractability and approximation effectiveness. By integrating these approximations into the probability estimation (PE) framework [Zhang et al., PRA 2018], we obtain significantly improved certified entropy bounds in the finite-size regime. We test our method on various bipartite and tripartite DI-QRNG protocols, using both simulated and experimental data. In all cases, it yields notably higher entropy rates with fewer device uses than the existing techniques. We further extend our analysis to the more demanding task of randomness amplification, demonstrating major performance gains without added complexity. These results offer an effective and ready-to-use method to prove security---with improved certified entropy rates---in the most common practical DI-QRNG protocols. Our algorithms and entropy certification with PE tools are publicly available under a non-commercial license at github. ► BibTeX data@article{Jee2026moreentropyfrom, doi = {10.22331/q-2026-03-10-2019}, url = {https://doi.org/10.22331/q-2026-03-10-2019}, title = {More entropy from shorter experiments using polytope approximations to the quantum set}, author = {Jee, Hyejung H. and Curchod, Florian J. and Almeida, Mafalda L.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2019}, month = mar, year = {2026} }► References [1] Masahito Hayashi and Toyohiro Tsurumaru. ``More efficient privacy amplification with less random seeds via dual universal hash function''. IEEE Transactions on Information Theory 62, 2213–2232 (2016). https://doi.org/10.1109/TIT.2016.2526018 [2] Cameron Foreman, Richie Yeung, Alec Edgington, and Florian J. Curchod. ``Cryptomite: A versatile and user-friendly library of randomness extractors''. Quantum 9, 1584 (2025). https://doi.org/10.22331/q-2025-01-08-1584 [3] Stefano Pironio, Antonio Acín, Serge Massar, A Boyer de La Giroday, Dzmitry N. Matsukevich, Peter Maunz, Steven Olmschenk, David Hayes, Le Luo, Timothy A. Manning, et al. ``Random numbers certified by Bell’s theorem''. Nature 464, 1021–1024 (2010). https://doi.org/10.1038/nature09008 [4] Stefano Pironio and Serge Massar. ``Security of practical private randomness generation''. Physical Review A 87, 012336 (2013). https://doi.org/10.1103/PhysRevA.87.012336 [5] Serge Fehr, Ran Gelles, and Christian Schaffner. ``Security and composability of randomness expansion from Bell inequalities''. Physical Review A 87, 012335 (2013). https://doi.org/10.1103/PhysRevA.87.012335 [6] Frédéric Dupuis, Omar Fawzi, and Renato Renner. ``Entropy accumulation''. Communications in Mathematical Physics 379, 867–913 (2020). https://doi.org/10.1007/s00220-020-03839-5 [7] Yanbao Zhang, Emanuel Knill, and Peter Bierhorst. ``Certifying quantum randomness by probability estimation''. Physical Review A 98, 040304 (2018). https://doi.org/10.1103/PhysRevA.98.040304 [8] Emanuel Knill, Yanbao Zhang, and Peter Bierhorst. ``Generation of quantum randomness by probability estimation with classical side information''.

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Brandão, Ravishankar Ramanathan, Andrzej Grudka, Karol Horodecki, Michał Horodecki, Paweł Horodecki, Tomasz Szarek, and Hanna Wojewódka. ``Realistic noise-tolerant randomness amplification using finite number of devices''. Nature Communications 7, 11345 (2016). https://doi.org/10.1038/ncomms11345Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-10 14:09:50: Could not fetch cited-by data for 10.22331/q-2026-03-10-2019 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-10 14:09:50: 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 introduce a systematic method for constructing polytope approximations to the quantum set in a variety of device-independent quantum random number generation (DI-QRNG) protocols. Our approach relies on two general-purpose algorithms that iteratively refine an initial outer-polytope approximation, guided by typical device behaviour and cryptographic intuition. These refinements strike a balance between computational tractability and approximation effectiveness. By integrating these approximations into the probability estimation (PE) framework [Zhang et al., PRA 2018], we obtain significantly improved certified entropy bounds in the finite-size regime. We test our method on various bipartite and tripartite DI-QRNG protocols, using both simulated and experimental data. In all cases, it yields notably higher entropy rates with fewer device uses than the existing techniques. We further extend our analysis to the more demanding task of randomness amplification, demonstrating major performance gains without added complexity. These results offer an effective and ready-to-use method to prove security---with improved certified entropy rates---in the most common practical DI-QRNG protocols. Our algorithms and entropy certification with PE tools are publicly available under a non-commercial license at github. ► BibTeX data@article{Jee2026moreentropyfrom, doi = {10.22331/q-2026-03-10-2019}, url = {https://doi.org/10.22331/q-2026-03-10-2019}, title = {More entropy from shorter experiments using polytope approximations to the quantum set}, author = {Jee, Hyejung H. and Curchod, Florian J. and Almeida, Mafalda L.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2019}, month = mar, year = {2026} }► References [1] Masahito Hayashi and Toyohiro Tsurumaru. ``More efficient privacy amplification with less random seeds via dual universal hash function''. IEEE Transactions on Information Theory 62, 2213–2232 (2016). https://doi.org/10.1109/TIT.2016.2526018 [2] Cameron Foreman, Richie Yeung, Alec Edgington, and Florian J. Curchod. ``Cryptomite: A versatile and user-friendly library of randomness extractors''. Quantum 9, 1584 (2025). https://doi.org/10.22331/q-2025-01-08-1584 [3] Stefano Pironio, Antonio Acín, Serge Massar, A Boyer de La Giroday, Dzmitry N. Matsukevich, Peter Maunz, Steven Olmschenk, David Hayes, Le Luo, Timothy A. Manning, et al. ``Random numbers certified by Bell’s theorem''. Nature 464, 1021–1024 (2010). https://doi.org/10.1038/nature09008 [4] Stefano Pironio and Serge Massar. ``Security of practical private randomness generation''. Physical Review A 87, 012336 (2013). https://doi.org/10.1103/PhysRevA.87.012336 [5] Serge Fehr, Ran Gelles, and Christian Schaffner. ``Security and composability of randomness expansion from Bell inequalities''. Physical Review A 87, 012335 (2013). https://doi.org/10.1103/PhysRevA.87.012335 [6] Frédéric Dupuis, Omar Fawzi, and Renato Renner. ``Entropy accumulation''. Communications in Mathematical Physics 379, 867–913 (2020). https://doi.org/10.1007/s00220-020-03839-5 [7] Yanbao Zhang, Emanuel Knill, and Peter Bierhorst. ``Certifying quantum randomness by probability estimation''. Physical Review A 98, 040304 (2018). https://doi.org/10.1103/PhysRevA.98.040304 [8] Emanuel Knill, Yanbao Zhang, and Peter Bierhorst. ``Generation of quantum randomness by probability estimation with classical side information''.

Hadronic Journal Supplement 8, 329–345 (1993). [17] Miguel Navascués, Stefano Pironio, and Antonio Acín. ``Bounding the set of quantum correlations''.

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More entropy from shorter experiments using polytope approximations to the quantum set

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