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Device-independent randomness expansion against quantum side information


The ability to produce random numbers that are unknown to any outside party is crucial for many applications. Device-independent randomness generation1,2,3,4 does not require trusted devices and therefore provides strong guarantees of the security of the output, but it comes at the price of requiring the violation of a Bell inequality for implementation. A further challenge is to make the bounds in the security proofs tight enough to allow randomness expansion with contemporary technology. Although randomness has been generated in recent experiments5,6,7,8,9, the amount of randomness consumed in doing so has been too high to certify expansion based on existing theory. Here we present an experiment that demonstrates device-independent randomness expansion1,2,3,10,11,12,13,14,15. By developing a Bell test setup with a single-photon detection efficiency of around 84% and by using a spot-checking protocol, we achieve a net gain of 2.57 × 108 certified bits with a soundness error of 3.09 × 10−12. The experiment ran for 19.2 h, which corresponds to an average rate of randomness generation of 13,527 bits per second. By developing the entropy accumulation theorem4,16,17, we establish security against quantum adversaries. We anticipate that this work will lead to further improvements that push device-independence towards commercial viability.

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Fig. 1: Conceptual sketch of the DIRNE protocol setup.
Fig. 2: Schematic of the experiment.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability

All relevant codes or algorithms are available from the corresponding authors upon reasonable request.


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We thank C.-L. Li for experimental assistance and J.-D. Bancal and E. Tan for comments on an earlier draft. This work was supported by the National Key R&D Program of China (grant nos. 2017YFA0303900 and 2017YFA0304000), the National Natural Science Foundation of China, the Chinese Academy of Sciences, the Shanghai Municipal Science and Technology Major Project (grant no. 2019SHZDZX01), the Anhui Initiative in Quantum Information Technologies, the Guangdong Innovative and Entrepreneurial Research Team Program (grant no. 2019ZT08X324), the Key Area R&D Program of Guangdong Province (grant no. 2020B0303010001), the Quantum Communications Hub of the Engineering and Physical Sciences Research Council (EPSRC) (grant nos. EP/M013472/1 and EP/T001011/1) and an EPSRC First Grant (grant no. EP/P016588/1). We are grateful for computational support from the University of York High Performance Computing service, Viking, which was used for the randomness extraction.

Author information

Authors and Affiliations



R.C., J.F., Q.Z. and J.-W.P. conceived the research. Y.L., J.F., Q.Z. and J.-W.P. designed the experiment. W.-Z.L., M.-H.L., S.-R.Z. and Y.L. designed and implemented the entangled photon pair source. W.-Z.L. designed the data acquisition software. B.B. and J.Z. designed the biased and unbiased quantum random number generators for measurement setting choices. S.R., P.J.B. and R.C. developed the theory. S.R., P.J.B., W.-Z.L. and R.C. performed the protocol analysis, numerical modelling and randomness extraction. All authors contributed to the experimental realization, data analysis and manuscript preparation.

Corresponding authors

Correspondence to Roger Colbeck, Jingyun Fan, Qiang Zhang or Jian-Wei Pan.

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The authors declare no competing interests.

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Peer review informationNature Physics thanks Thomas Vidick 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.

Extended data

Extended Data Fig. 1 Rounds needed and expansion rate for the main and space-like experiments.

a, We estimate the minimum number of experimental runs with our revised EAT theory to witness randomness expansion as a function of CHSH violation (smooth curve) with a soundness error 3.09 × 10−12. The red square, yellow circle and green cross indicate the previous8, space-like and main experimental conditions, respectively. b, We estimate the randomness expansion rate based on our revised EAT theory as a function of number of rounds (smooth line) and the asymptotic rate (dashed line) with a soundness error 3.09 × 10−12. The cross and circle indicate the experimental parameters used, red indicates the main experiment and blue indicates the space-like experiment.

Source data

Supplementary information

Supplementary Information

Supplementary Figs. 1–5, discussion and Tables 1–7.

Source data

Source Data Extended Data Fig. 1

Source data for Extended Data Fig. 1. In a, the first column always represents the CHSH score and the second column represents the number of rounds, n. In b, the first and the third columns alway represent the number of rounds n for main setup and space-like setup, respectively. The second and the fourth columns alway represent the rate of randomness expansion for main setup and space-like setup, respectively.

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Liu, WZ., Li, MH., Ragy, S. et al. Device-independent randomness expansion against quantum side information. Nat. Phys. 17, 448–451 (2021).

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