Random numbers certified by Bell’s theorem


Randomness is a fundamental feature of nature and a valuable resource for applications ranging from cryptography and gambling to numerical simulation of physical and biological systems. Random numbers, however, are difficult to characterize mathematically1, and their generation must rely on an unpredictable physical process2,3,4,5,6. Inaccuracies in the theoretical modelling of such processes or failures of the devices, possibly due to adversarial attacks, limit the reliability of random number generators in ways that are difficult to control and detect. Here, inspired by earlier work on non-locality-based7,8,9 and device-independent10,11,12,13,14 quantum information processing, we show that the non-local correlations of entangled quantum particles can be used to certify the presence of genuine randomness. It is thereby possible to design a cryptographically secure random number generator that does not require any assumption about the internal working of the device. Such a strong form of randomness generation is impossible classically and possible in quantum systems only if certified by a Bell inequality violation15. We carry out a proof-of-concept demonstration of this proposal in a system of two entangled atoms separated by approximately one metre. The observed Bell inequality violation, featuring near perfect detection efficiency, guarantees that 42 new random numbers are generated with 99 per cent confidence. Our results lay the groundwork for future device-independent quantum information experiments and for addressing fundamental issues raised by the intrinsic randomness of quantum theory.

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Figure 1: Experimental realization of private random number generator using two 171 Yb + qubits trapped in independent vacuum chambers.
Figure 2: Plot of the function f(I ) bounding the output randomness.
Figure 3: Bound nf(I ) on the minimum entropy produced versus the number of trials n for an observed CHSH violation of , and a confidence level 1 -  δ = 99%.


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We thank R. Colbeck for sharing his PhD thesis with us. This work was supported by the Swiss NCCR Quantum Photonics, the European ERC-AG QORE, the European projects QAP and COM-PAS, the ERC starting grant PERCENT, the Spanish MEC FIS2007-60182 and Consolider-Ingenio QOIT projects, Generalitat de Catalunya, Caixa Manresa, Fundacio Cellex Barcelona, the Interuniversity Attraction Poles Photonics@be Programme (Belgian Science Policy), the Brussels-Capital Region through the project CRYPTASC and a BB2B Grant, the US Army Research Office with funds from IARPA, the National Science Foundation (NSF) Physics at the Information Frontier Program, and the NSF Physics Frontier Center at JQI.

Author Contributions S.P., A.A., S.M. and A.B.d.l.G. developed the theoretical aspects of this work. D.N.M., P.M., S.O., D.H., L.L., T.A.M. and C.M. performed the experiments.

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Correspondence to A. Acín.

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Supplementary Information

This Supplementary Information file comprises: A Theoretical results; B Quantum randomness expanders, C Requirements on the devices and D Experiment. It also contains Supplementary Table 2 and Supplementary References. (PDF 245 kb)

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Pironio, S., Acín, A., Massar, S. et al. Random numbers certified by Bell’s theorem. Nature 464, 1021–1024 (2010). https://doi.org/10.1038/nature09008

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