Maximal device-independent randomness in every dimension

Nature Physics (2026)Cite this article Many scientific and security protocols rely on sources of unpredictable and private random numbers. Device-independent quantum random number generation is a framework that makes use of the intrinsic randomness of quantum processes to generate numbers that are fundamentally unpredictable according to our current understanding of physics. However, the difficulty of controlling quantum systems makes it challenging to carry out device-independent protocols in practice. It is, therefore, desirable to harness the full power of the quantum degrees of freedom that one can control. It is known that no more than 2 log(d) bits of private device-independent randomness can be extracted from a quantum system of local dimension d. Here we demonstrate that this bound can be achieved for all d by providing a family of explicit protocols. 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J.V. was supported by the NSF (Grant No. DMS-2348720). We acknowledge the QuantERA II Programme (Grant Agreement No. 101017733, VERIqTAS).Department of Mathematics, University of York, York, UKMáté FarkasDepartment of Mathematics, University of Auckland, Auckland, New ZealandJurij VolčičQMATH, Department of Mathematical Sciences, University of Copenhagen, Copenhagen, DenmarkSigurd A. L. Storgaard & Laura MančinskaQuantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area, Shenzhen, ChinaRanyiliu ChenSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarM.F., J.V., S.A.L.S., R.C. and L.M. conceived the idea, crafted the proofs and prepared the paper.Correspondence to Máté Farkas.The authors declare no competing interests.Nature Physics thanks Yanbao Zhang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Technical proofs.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsFarkas, M., Volčič, J., Storgaard, S.A.L. et al. Maximal device-independent randomness in every dimension. Nat. Phys. (2026). https://doi.org/10.1038/s41567-025-03141-yDownload citationReceived: 20 March 2025Accepted: 20 November 2025Published: 13 January 2026Version of record: 13 January 2026DOI: https://doi.org/10.1038/s41567-025-03141-yAnyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative