Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Quantum networks self-test all entangled states

Nature Physics volume 19, pages 670–675 (2023)Cite this article

Subjects

Abstract

Certifying quantum properties with minimal assumptions is a fundamental problem in quantum information science. Self-testing is a method to infer the underlying physics of a quantum experiment only from the measured statistics1,2. Although all bipartite pure entangled states can be self-tested3, little is known about how to self-test quantum states of an arbitrary number of systems. Here we introduce a framework of network-assisted self-testing and use it to self-test any pure entangled quantum state of an arbitrary number of systems. The scheme requires the preparation of a number of singlets that scales linearly with the number of systems, and the implementation of standard projective and Bell measurements, all feasible with current technology4. When all the network constraints are exploited, the obtained self-testing certification is stronger than what is achievable in any Bell-type scenario. Our work shows how properly designed networks offer new opportunities for the certification of quantum phenomena.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Network-assisted self-testing in a scenario for M = 5 devices to self-test a three-party state \({\left\vert \psi \right\rangle}\).
Fig. 2: The network scenarios used to self-test any pure state \({\left\vert {\psi }_{N}\right\rangle}\).
Fig. 3: The ingredients required for our self-testing protocols.

Similar content being viewed by others

Data availability

This work does not have any associated data.

References

  1. Mayers, D. & Yao, A. Self testing quantum apparatus. Quantum Info. Comput. 4, 273–286 (2004).

    MathSciNet  MATH  Google Scholar 

  2. Šupić, I. & Bowles, J. Self-testing of quantum systems: a review. Quantum 4, 337 (2020).

    Article  Google Scholar 

  3. Coladangelo, A., Goh, K. T. & Scarani, V. All pure bipartite entangled states can be self-tested. Nat. Commun. 8, 15485 (2017).

    Article  ADS  Google Scholar 

  4. Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: a vision for the road ahead. Science 362, eaam9288 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964).

    Article  MathSciNet  Google Scholar 

  6. Eisert, J. et al. Quantum certification and benchmarking. Nat. Rev. Phys. 2, 382–390 (2020).

    Article  Google Scholar 

  7. Fano, U. Description of states in quantum mechanics by density matrix and operator techniques. Rev. Mod. Phys. 29, 74–93 (1957).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Vogel, K. & Risken, H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 2847–2849 (1989).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. Pironio, S. et al. Device-independent quantum key distribution secure against collective attacks. New J. Phys. 11, 045021 (2009).

  11. Colbeck, R. Quantum and Relativistic Protocols for Secure Multi-Party Computation. PhD thesis, Univ. Cambridge (2006); https://arxiv.org/abs/0911.3814

  12. Mayers, D. & Yao, A. Quantum cryptography with imperfect apparatus. In Proc. 39th Annual Symposium on Foundations of Computer Science (cat. no. 98CB36280) 503–509 (IEEE, 1998); https://doi.org/10.1109/sfcs.1998.743501

  13. Summers, S. J. & Werner, R. Maximal violation of Bell’s inequalities is generic in quantum field theory. Commun. Math. Phys. 110, 247–259 (1987).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. McKague, M. Self-testing graph states. In Revised Selected Papers of the 6th Conference on Theory of Quantum Computation, Communication, and Cryptography Vol. 6745 (eds Bacon, D. et al.) 104–120 (Springer, 2011); https://doi.org/10.1007/978-3-642-54429-3_7

  15. Wu, X. et al. Robust self-testing of the three-qubit W state. Phys. Rev. A 90, 042339 (2014).

    Article  ADS  Google Scholar 

  16. Pál, K. F., Vértesi, T. & Navascués, M. Device-independent tomography of multipartite quantum states. Phys. Rev. A 90, 042340 (2014).

    Article  ADS  Google Scholar 

  17. Šupić, I., Coladangelo, A., Augusiak, R. & Acín, A. Self-testing multipartite entangled states through projections onto two systems. N. J. Phys. 20, 083041 (2018).

    Article  Google Scholar 

  18. Baccari, F., Augusiak, R., Šupić, I., Tura, J. & Acín, A. Scalable Bell inequalities for qubit graph states and robust self-testing. Phys. Rev. Lett. 124, 020402 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  19. Buscemi, F. All entangled quantum states are nonlocal. Phys. Rev. Lett. 108, 200401 (2012).

    Article  ADS  Google Scholar 

  20. Branciard, C., Rosset, D., Liang, Y.-C. & Gisin, N. Measurement-device-independent entanglement witnesses for all entangled quantum states. Phys. Rev. Lett. 110, 060405 (2013).

    Article  ADS  Google Scholar 

  21. Šupić, I., Hoban, M. J., Colomer, L. D. & Acín, A. Self-testing and certification using trusted quantum inputs. New J. Phys. 22, 073006 (2020).

  22. Bennett, C. H. et al. Remote state preparation. Phys. Rev. Lett. 87, 077902 (2001).

    Article  ADS  Google Scholar 

  23. Berry, D. W. & Sanders, B. C. Optimal remote state preparation. Phys. Rev. Lett. 90, 057901 (2003).

    Article  ADS  Google Scholar 

  24. Bancal, J.-D., Redeker, K., Sekatski, P., Rosenfeld, W. & Sangouard, N. Self-testing with finite statistics enabling the certification of a quantum network link. Quantum 5, 401 (2021).

    Article  Google Scholar 

  25. Gočanin, A., Šupić, I. & Dakić, B. Sample-efficient device-independent quantum state verification and certification. PRX Quantum 3, 010317 (2022).

    Article  ADS  Google Scholar 

  26. Coladangelo, A. Parallel self-testing of (tilted) EPR pairs via copies of (tilted) CHSH and the magic square game. Quantum Inf. Comput. 17, 831–865 (2017).

    MathSciNet  Google Scholar 

  27. 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  MATH  Google Scholar 

  28. Bowles, J., Šupić, I., Cavalcanti, D. & Acín, A. Self-testing of Pauli observables for device-independent entanglement certification. Phys. Rev. A 98, 042336 (2018).

  29. Wu, X., Bancal, J.-D., McKague, M. & Scarani, V. Device-independent parallel self-testing of two singlets. Phys. Rev. A 93, 062121 (2016).

    Article  ADS  Google Scholar 

  30. Coudron, M. & Natarajan, A. The parallel-repeated magic square game is rigid. Preprint at https://arxiv.org/abs/1609.06306 (2016).

  31. McKague, M. Self-testing in parallel with CHSH. Quantum 1, 1 (2017).

    Article  Google Scholar 

  32. Natarajan, A. & Vidick, T. A quantum linearity test for robustly verifying entanglement. In Proc. 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 1003–1015 (Association for Computing Machinery, 2017); https://doi.org/10.1145/3055399.3055468

  33. Branciard, C., Gisin, N. & Pironio, S. Characterizing the nonlocal correlations created via entanglement swapping. Phys. Rev. Lett. 104, 170401 (2010).

    Article  ADS  Google Scholar 

  34. Fritz, T. Beyond Bell’s Theorem II: scenarios with arbitrary causal structure. Commun. Math. Phys. 341, 391–434 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Renou, M.-O. et al. Genuine quantum nonlocality in the triangle network. Phys. Rev. Lett. 123, 140401 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  36. Šupić, I., Bancal, J.-D. & Brunner, N. Quantum nonlocality in networks can be demonstrated with an arbitrarily small level of independence between the sources. Phys. Rev. Lett. 125, 240403 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  37. Tavakoli, A., Pozas-Kerstjens, A., Luo, M.-X. & Renou, M.-O. Bell nonlocality in networks. Rep. Prog. Phys. 85, 056001 (2022).

  38. Renou, M.-O. et al. Quantum theory based on real numbers can be experimentally falsified. Nature 600, 625–629 (2021).

    Article  ADS  Google Scholar 

  39. Bădescu, C., O’Donnell, R. & Wright, J. Quantum state certification. In Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, 503–514 (Association for Computing Machinery, 2019); https://doi.org/10.1145/3313276.3316344

  40. Bancal, J.-D., Sangouard, N. & Sekatski, P. Noise-resistant device-independent certification of Bell state measurements. Phys. Rev. Lett. 121, 250506 (2018).

    Article  ADS  Google Scholar 

  41. Renou, M.-O., Kaniewski, J. & Brunner, N. Self-testing entangled measurements in quantum networks. Phys. Rev. Lett. 121, 250507 (2018).

    Article  ADS  Google Scholar 

  42. Morris, J., Saggio, V., Gočanin, A. & Dakić, B. Quantum verification and estimation with few copies. Adv. Quantum Technol. 5, 2100118 (2022).

    Article  Google Scholar 

  43. Yang, T. H., Vértesi, T., Bancal, J.-D., Scarani, V. & Navascués, M. Robust and versatile black-box certification of quantum devices. Phys. Rev. Lett. 113, 040401 (2014).

    Article  ADS  Google Scholar 

  44. Reichardt, B. W., Unger, F. & Vazirani, U. Classical command of quantum systems. Nature 496, 456–460 (2013).

    Article  ADS  Google Scholar 

  45. Coladangelo, A., Grilo, A. B., Jeffery, S. & Vidick, T. Verifier-on-a-leash: new schemes for verifiable delegated quantum computation, with quasilinear resources. In Advances in Cryptology – EUROCRYPT 2019 (eds Ishai, Y. & Rijmen, V.) 247–277 (Springer, 2019).

  46. McKague, M. Interactive proofs for BQP via self-tested graph states. Theory Comput. 12, 1–42 (2016).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was funded by the ERC through grants QUSCO (I.Š.) and AdG CERQUTE, (A.A.), grant PCI2021-122022-2B financed by MCIN/AEI/10.13039/501100011033 (M.-O.R.), the European Union Next Generation EU/PRTR (M.-O.R.), a Swiss National Fund Early Mobility Grant P2GEP2_19144 (M.-O.R.), a FQXi large grant FQXi-RFP-1803B (M.J.H.), the Government of Spain through grants FIS2020-TRANQI (A.A.), Retos Quspin (A.A.), NexGen Funds (A.A.) and Severo Ochoa CEX2019-000910-S (A.A.), Fundacio Cellex (A.A.), Fundacio Mir-Puig, (A.A.), Generalitat de Catalunya through grants CERCA (A.A.), the QuantERA II Programme through grant Veriqtas (A.A.), and QuantumCAT (A.A.), and the AXA Chair in Quantum Information Science (A.A.).

Author information

Authors and Affiliations

  1. CNRS, LIP6, Sorbonne Université, Paris, France

    Ivan Šupić

  2. ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona, Spain

    Joseph Bowles, Marc-Olivier Renou & Antonio Acín

  3. ICREA – Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain

    Antonio Acín

  4. Quantinuum, Oxford, UK

    Matty J. Hoban

Authors
  1. Ivan Šupić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph Bowles
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marc-Olivier Renou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Antonio Acín
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Matty J. Hoban
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to Matty J. Hoban.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Anna Pappa, Arthur Mehta and Petros Wallden for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Discussion and Figs. 1–3.

Rights and permissions

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 permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Šupić, I., Bowles, J., Renou, MO. et al. Quantum networks self-test all entangled states. Nat. Phys. 19, 670–675 (2023). https://doi.org/10.1038/s41567-023-01945-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-023-01945-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing