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Experimental randomness amplification | Nature

  • Colbeck, R. & Renner, R. Free randomness can be amplified. Nat. Phys. 8, 450–454 (2012).

    Article 
    CAS 

    Google Scholar
     

  • Hensen, B. et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature 526, 682–686 (2015).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Giustina, M. et al. Significant-loophole-free test of Bell’s theorem with entangled photons. Phys. Rev. Lett. 115, 250401 (2015).

    Article 
    ADS 
    PubMed 

    Google Scholar
     

  • Shalm, L. K. et al. Strong loophole-free test of local realism. Phys. Rev. Lett. 115, 250402 (2015).

    Article 
    ADS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Santha, M. & Vazirani, U. V. Generating quasi-random sequences from semi-random sources. J. Comput. Syst. Sci. 33, 75–87 (1986).

    Article 

    Google Scholar
     

  • Brown, G. W. History of RAND’s Random Digits: Summary Technical Report P113 (RAND, 1949); http://www.rand.org/pubs/papers/P113.html.

  • von Neumann, J. John von Neumann Collected Works, Vol. 5: Design of Computers, Theory of Automata and Numerical Analysis 768–770 (Pergamon Press, 1961).

  • Colbeck, R. & Renner, R. No extension of quantum theory can have improved predictive power. Nat. Commun. 2, 411 (2011).

    Article 
    ADS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Petrov, M. et al. Independent quality assessment of a commercial quantum random number generator. EPJ Quantum Technol. 9, 17 (2022).

    Article 
    ADS 

    Google Scholar
     

  • Rivest, R. L., Shamir, A. & Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21, 120–126 (1978).

    Article 
    MathSciNet 

    Google Scholar
     

  • Heninger, N., Durumeric, Z., Wustrow, E. & Halderman, J. A. Mining your Ps and Qs: detection of widespread weak keys in network devices. In Proc. 21st USENIX Security Symposium 205–220 (USENIX Association, 2012).

  • Lenstra, A. K. et al. Ron was wrong, Whit is right. IACR Cryptology ePrint Archive https://eprint.iacr.org/2012/064 (2012).

  • RAND Coorporation A Million Random Digits with 100,000 Normal Deviates (American Book Publishers, 2001).

  • Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964).

    Article 
    MathSciNet 

    Google Scholar
     

  • Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969).

    Article 
    ADS 

    Google Scholar
     

  • Freedman, S. J. & Clauser, J. F. Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938–941 (1972).

    Article 
    ADS 
    CAS 

    Google Scholar
     

  • Aspect, A., Grangier, P. & Roger, G. Experimental realization of Einstein–Podolsky–Rosen–Bohm gedankenexperiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 49, 91–94 (1982).

    Article 
    ADS 

    Google Scholar
     

  • Rowe, M. A. et al. Experimental violation of a Bell’s inequality with efficient detection. Nature 409, 791–794 (2001).

  • Salart, D., Baas, A., Branciard, C., Gisin, N. & Zbinden, H. Testing the speed of ‘spooky action at a distance’. Nature 454, 861–864 (2008).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Kessler, M. & Arnon-Friedman, R. Device-independent randomness amplification and privatization. IEEE J. Sel. Areas Inf. Theory 1, 568–584 (2020).

    Article 

    Google Scholar
     

  • Vazirani, U. Strong communication complexity or generating quasirandom sequences form two communicating semi-random sources. Combinatorica 7, 375–392 (1987).

    Article 
    MathSciNet 

    Google Scholar
     

  • Chor, B. & Goldreich, O. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17, 230–261 (1988).

    Article 
    MathSciNet 

    Google Scholar
     

  • Liu, M. et al. Certified randomness using a trapped-ion quantum processor. Nature 640, 343–348 (2025).

    Article 
    ADS 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Blais, A., Grimsmo, A. L., Girvin, S. M. & Wallraff, A. Circuit quantum electrodynamics. Rev. Mod. Phys. 93, 025005 (2021).

    Article 
    ADS 
    MathSciNet 
    CAS 

    Google Scholar
     

  • Storz, S. et al. Loophole-free Bell inequality violation with superconducting circuits. Nature 617, 265–270 (2023).

    Article 
    ADS 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Portmann, C. & Renner, R. Security in quantum cryptography. Rev. Mod. Phys. 94, 025008 (2022).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Pütz, G., Rosset, D., Barnea, T. J., Liang, Y.-C. & Gisin, N. Arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality. Phys. Rev. Lett. 113, 190402 (2014).

    Article 
    ADS 
    PubMed 

    Google Scholar
     

  • Sandfuchs, M., Ferradini, C. & Renner, R. Randomness from causally independent processes. Preprint at https://arxiv.org/abs/2510.05203 (2025).

  • Abellán, C., Amaya, W., Mitrani, D., Pruneri, V. & Mitchell, M. W. Generation of fresh and pure random numbers for loophole-free Bell tests. Phys. Rev. Lett. 115, 250403 (2015).

    Article 
    ADS 
    PubMed 

    Google Scholar
     

  • Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).

    Article 
    ADS 

    Google Scholar
     

  • Kurpiers, P. et al. Deterministic quantum state transfer and remote entanglement using microwave photons. Nature 558, 264–267 (2018).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Kurpiers, P., Walter, T., Magnard, P., Salathe, Y. & Wallraff, A. Characterizing the attenuation of coaxial and rectangular microwave-frequency waveguides at cryogenic temperatures. EPJ Quantum Technol. 4, 8 (2017).

    Article 
    ADS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Magnard, P. et al. Microwave quantum link between superconducting circuits housed in spatially separated cryogenic systems. Phys. Rev. Lett. 125, 260502 (2020).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Walter, T. et al. Rapid high-fidelity single-shot dispersive readout of superconducting qubits. Phys. Rev. Appl. 7, 054020 (2017).

    Article 
    ADS 

    Google Scholar
     

  • Magnard, P. et al. Fast and unconditional all-microwave reset of a superconducting qubit. Phys. Rev. Lett. 121, 060502 (2018).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Cirac, J. I., Zoller, P., Kimble, H. J. & Mabuchi, H. Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett. 78, 3221–3224 (1997).

    Article 
    ADS 
    CAS 

    Google Scholar
     

  • Storz, S. et al. Complete self-testing of a system of remote superconducting qubits. Phys. Rev. Lett. 135, 030801 (2025).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Garg, A. & Mermin, N. D. Detector inefficiencies in the Einstein–Podolsky–Rosen experiment. Phys. Rev. D 35, 3831–3835 (1987).

    Article 
    ADS 
    CAS 

    Google Scholar
     

  • Eberhard, P. H. Background level and counter efficiencies required for a loophole-free Einstein–Podolsky–Rosen experiment. Phys. Rev. A 47, R747–R750 (1993).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Larsson, J.-A. Loopholes in Bell inequality tests of local realism. J. Phys. A 47, 424003 (2014).

    Article 
    MathSciNet 

    Google Scholar
     

  • Rukhin, A. et al. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications rev.1a (NIST, 2010); https://csrc.nist.gov/projects/random-bit-generation/documentation-and-software.

  • Marsaglia, G. & Tsang, W. W. Some difficult-to-pass tests of randomness. J. Stat. Softw. https://doi.org/10.18637/jss.v007.i03 (2022).

  • Kelsey, J., Brandão, L. T., Peralta, R. & Booth, H. A Reference for Randomness Beacons: Format and Protocol Version 2 Technical Report (NIST, 2019); https://doi.org/10.6028/NIST.IR.8213-draft.

  • Kavuri, G. A. et al. Traceable random numbers from a non-local quantum advantage. Nature 642, 916–921 (2025).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Yao, A. C. Protocols for secure computations. In 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982) 160–164 (IEEE, 1982).

  • Goldreich, O., Micali, S. & Wigderson, A. How to play any mental game. In STOC ’87: Proc. Nineteenth Annual ACM Symposium on Theory of Computing 218–229 (ACM, 1987); https://doi.org/10.1145/28395.28420.

  • Dodis, Y., Ong, S. J., Prabhakaran, M. & Sahai, A. On the (im)possibility of cryptography with imperfect randomness. In 45th Annual IEEE Symposium on Foundations of Computer Science 196–205 (IEEE, 2004).

  • Nadlinger, D. P. et al. Experimental quantum key distribution certified by Bell’s theorem. Nature 607, 682–686 (2022).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Zhang, W. et al. A device-independent quantum key distribution system for distant users. Nature 607, 687–691 (2022).

    Article 
    ADS 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Ekert, A. & Renner, R. The ultimate physical limits of privacy. Nature 507, 443–447 (2014).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Li, M.-H. et al. Test of local realism into the past without detection and locality loopholes. Phys. Rev. Lett. 121, 080404 (2018).

    Article 
    ADS 
    PubMed 

    Google Scholar
     

  • Liu, Y. et al. Device-independent quantum random-number generation. Nature 562, 548–551 (2018).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Zhang, Y. et al. Experimental low-latency device-independent quantum randomness. Phys. Rev. Lett. 124, 010505 (2020).

    Article 
    ADS 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Shalm, L. K. et al. Device-independent randomness expansion with entangled photons. Nat. Phys. 17, 452–456 (2021).

    Article 
    CAS 

    Google Scholar
     

  • Li, M.-H. et al. Experimental realization of device-independent quantum randomness expansion. Phys. Rev. Lett. 126, 050503 (2021).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Liu, W.-Z. et al. Device-independent randomness expansion against quantum side information. Nat. Phys. https://doi.org/10.1038/s41567-020-01147-2 (2021).

  • Bierhorst, P. et al. Experimentally generated randomness certified by the impossibility of superluminal signals. Nature 556, 223–226 (2018).

    Article 
    ADS 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Liu, W.-Z. et al. Toward a photonic demonstration of device-independent quantum key distribution. Phys. Rev. Lett. 129, 050502 (2022).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Hensen, B. et al. Loophole-free Bell test using electron spins in diamond: second experiment and additional analysis. Sci. Rep. 6, 30289 (2016).

    Article 
    ADS 
    CAS 
    PubMed 
    PubMed Central 

    Google Scholar
     

  • Rosenfeld, W. et al. Event-ready Bell test using entangled atoms simultaneously closing detection and locality loopholes. Phys. Rev. Lett. 119, 010402 (2017).

    Article 
    ADS 
    PubMed 

    Google Scholar
     

  • Frauchiger, D., Renner, R. & Troyer, M. True randomness from realistic quantum devices. Preprint at https://arxiv.org/abs/1311.4547 (2013).

  • Portmann, C. & Renner, R. Cryptographic security of quantum key distribution. Preprint at https://arxiv.org/abs/1409.3525 (2014).

  • Ferradini, C., Sandfuchs, M., Wolf, R. & Renner, R. Defining security in quantum key distribution. Preprint at https://arxiv.org/abs/2509.13405 (2025).

  • Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. & Wehner, S. Bell nonlocality. Rev. Mod. Phys. 86, 419–478 (2014).

    Article 
    ADS 

    Google Scholar
     

  • Ekert, A. K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991).

    Article 
    ADS 
    MathSciNet 
    CAS 
    PubMed 

    Google Scholar
     

  • Acín, A. et al. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007).

    Article 
    ADS 
    PubMed 

    Google Scholar
     

  • Colbeck, R. Quantum And Relativistic Protocols For Secure Multi-Party Computation. PhD thesis, Univ. Cambridge (2009).

  • Pironio, S. et al. Random numbers certified by Bell’s theorem. Nature 464, 1021–1024 (2010).

    Article 
    ADS 
    CAS 
    PubMed 

    Google Scholar
     

  • Renner, R. Security of Quantum Key Distribution. PhD thesis, ETH Zurich (2005); https://doi.org/10.3929/ethz-a-005115027.

  • Tomamichel, M. & Hayashi, M. A hierarchy of information quantities for finite block length analysis of quantum tasks. IEEE Trans. Inf. Theory 59, 7693–7710 (2013).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Arnon-Friedman, R., Portmann, C. & Scholz, V. B. Quantum-proof multi-source randomness extractors in the Markov model. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016), Leibniz International Proceedings in Informatics (LIPIcs) Vol. 61 (ed. Broadbent, A.) 2:1–2:34 (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2016).

  • Dupuis, F., Fawzi, O. & Renner, R. Entropy accumulation. Commun. Math. Phys. 379, 867–913 (2020).

    Article 
    ADS 
    MathSciNet 

    Google Scholar
     

  • Metger, T., Fawzi, O., Sutter, D. & Renner, R. Generalised entropy accumulation. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) 844–850 (IEEE, 2022).

  • Storz, S. Non-local Physics with Superconducting Circuits. PhD thesis, ETH Zurich (2023).

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