Expanding bipartite Bell inequalities for maximum multi-partite randomness

Lewis Wooltorton; Peter Brown; Roger Colbeck

doi:10.22331/q-2025-12-05-1930

Expanding bipartite Bell inequalities for maximum multi-partite randomness

Lewis Wooltorton^1,2,3, Peter Brown⁴, and Roger Colbeck^5,1

¹Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
²Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical & Electronic Engineering, University of Bristol, Bristol BS8 1FD, United Kingdom
³Inria, ENS de Lyon, LIP, 46 Allee d’Italie, 69364 Lyon Cedex 07, France
⁴Télécom Paris, LTCI, Institut Polytechnique de Paris, 19 Place Marguerite Perey, 91120 Palaiseau, France
⁵Department of Mathematics, King's College London, Strand, London, WC2R 2LS, United Kingdom

Published:	2025-12-05, volume 9, page 1930
Editor:	Remigiusz Augusiak
Eprint:	arXiv:2308.07030v3
Doi:	https://doi.org/10.22331/q-2025-12-05-1930
Citation:	Quantum 9, 1930 (2025).

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Abstract

Nonlocal tests on multi-partite quantum correlations form the basis of protocols that certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate test difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, but achieving their maximum violation places constraints on the underlying quantum system, which can reduce the rate of randomness generation. As a result there is often a trade-off between maximum randomness and the amount of violation of a given Bell inequality. Here, we explore this trade-off for more than two parties. More precisely, we study the maximum amount of randomness that can be certified by correlations with a particular violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. For any even number of parties, we find that maximum randomness cannot occur beyond a threshold quantum violation, which increases with the number of parties, and we give a conjectured form of the maximum randomness in terms of the MABK value. We also show that maximum randomness can be obtained for any MABK violation for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a technique for randomness certification, which we call "expanding Bell inequalities''. Our technique allows a bipartite Bell expression to be used as a seed, and transformed into a multi-partite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.

Featured image: Our technique uses a Bell inequality tailored to randomness certification between two parties, and expands it into a Bell inequality that certifies maximum randomness from all parties.

Popular summary

Sources of genuine private randomness have numerous applications in modern technology, and quantum processes are an excellent candidate: the numbers they generate are provably uncorrelated with any adversary according to the laws of physics. Unfortunately, this often requires a detailed description of the devices used, which can be very challenging to achieve in practice. Furthermore, any aspect improperly characterized could be exploited by an adversary, opening up security loopholes. A device-independent (DI) random number generator circumvents this issue. In this setting, security is proven from experimental observations, called correlations, and knowledge of the underlying quantum process is not required.

In this work, we identify optimal correlations for DI randomness generation between many non-communicating parties, and show how to certify them using a single non-classical witness, called a Bell inequality. Specifically, we introduce a technique that uses a Bell inequality tailored to randomness certification between two parties, and expands it into a Bell inequality that certifies maximum randomness from all parties. We then apply this technique to show that maximum randomness generation is incompatible with highly non-classical correlations in the multi-partite setting, and that this incompatibility vanishes in the limit of infinitely many parties. Our results provide foundational insights into the relationship between non-classicality and DI randomness, whilst simultaneously introducing new protocols. We also envisage our techniques being used to find further applications in DI cryptography.

► BibTeX data

@article{Wooltorton2025expandingbipartite,
  doi = {10.22331/q-2025-12-05-1930},
  url = {https://doi.org/10.22331/q-2025-12-05-1930},
  title = {Expanding bipartite {B}ell inequalities for maximum multi-partite randomness},
  author = {Wooltorton, Lewis and Brown, Peter and Colbeck, Roger},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1930},
  month = dec,
  year = {2025}
}

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Cited by

[1] Bingzi Huo, Kunkun Wang, Dengke Qu, Junjing Xing, Xiang Zhan, Xiaojian Huang, Huixia Gao, Lei Xiao, and Peng Xue, "Sequential Sharing of Quantum Nonlocality via Projective Measurements", Physical Review Letters 136 10, 100201 (2026).

[2] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Device-Independent Quantum Key Distribution with Arbitrarily Small Nonlocality", Physical Review Letters 132 21, 210802 (2024).

[3] Maria Ciudad Alañón, Daniel Centeno, Andrew Watford, and Elie Wolfe, "Certifying Randomness or its Lack Thereof for General Network Scenarios", arXiv:2510.20993, (2025).

[4] Emanuele Polino, Luis Villegas-Aguilar, Davide Poderini, Nathan Walk, Farzad Ghafari, Marco Túlio Quintino, Alexey Lyasota, Sven Rogge, Rafael Chaves, Geoff J. Pryde, Eric G. Cavalcanti, Nora Tischler, and Sergei Slussarenko, "Experimental quantum randomness enhanced by a quantum network", arXiv:2412.16973, (2024).

[5] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Genuine multipartite entanglement is not necessary for standard device-independent conference key agreement", arXiv:2503.21290, (2025).

[6] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Genuine Multipartite Entanglement is Not Necessary for Standard Device-Independent Conference Key Agreement", Physical Review Letters 135 22, 220803 (2025).

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