- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review A

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Robust self-testing and certified randomness based on an arbitrary-input Bell inequality

Rajdeep Paul*, Sneha Munshi†, and A. K. Pan‡

  • *Contact author: rajdeeppaul22@gmail.com
  • †Contact author: snehamunshi5@gmail.com
  • ‡Contact author: akp@phy.iith.ac.in
PDF

Phys. Rev. A 113, 042211 – Published 14 April, 2026

DOI: https://doi.org/10.1103/z4tp-mc5z

Abstract

Self-testing is the strongest certification procedure that uniquely characterizes the physical system based on the observed statistics, without any knowledge of the inner workings of the devices. The optimal quantum violation of a Bell inequality enables such a device-independent (DI) self-testing of the source and the measurement devices. In this work, we demonstrate the DI self-testing based on the arbitrary-input chained Bell inequality. We devise a systematic and elegant sum-of-squares technique enabling dimension-independent optimization of the quantum violation. Our approach enables the derivation of the state along with the relationship between the local observables directly from the optimization condition. One significant aspect is the robustness of such self-testing in real experimental situations involving noise and imperfection, leading to deviation from the optimal quantum violation. We provide an analytical technique for robust self-testing in the presence of noise. As an application of our scheme, we demonstrate the generation of two-bit DI randomness and analyze the robustness of such randomness. Our optimization method is both simple and elegant, making it suitable for deriving the optimal quantum violation of various arbitrary-input Bell inequalities.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (64)

  1. J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  2. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
  3. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  4. S. L. Braunstein and C. M. Caves, Wringing out better Bell inequalities, Ann. Phys. (NY) 202, 22 (1990).
  5. A. Acín, S. Massar, and S. Pironio, Randomness versus nonlocality and entanglement, Phys. Rev. Lett. 108, 100402 (2012).
  6. R. Colbeck and A. Kent, Private randomness expansion with untrusted devices, J. Phys. A: Math. Theor. 44, 095305 (2011).
  7. L. Wooltorton, P. Brown, and R. Colbeck, Device-independent quantum key distribution with arbitrarily small nonlocality, Phys. Rev. Lett. 132, 210802 (2024).
  8. S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, Device-independent quantum key distribution secure against collective attacks, New J. Phys. 11, 045021 (2009).
  9. M. Farkas, M. Balanzó-Juandó, K. Łukanowski, J. Kołodyński, and A. Acín, Bell nonlocality is not sufficient for the security of standard device-independent quantum key distribution protocols, Phys. Rev. Lett. 127, 050503 (2021).
  10. A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-independent security of quantum cryptography against collective attacks, Phys. Rev. Lett. 98, 230501 (2007).
  11. D. Mayers and A. Yao, Quantum cryptography with imperfect apparatus, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science, FOCS '98 (IEEE Computer Society, USA, 1998), p. 503.
  12. Dominic Mayers and Andrew Yao, Self testing quantum apparatus, arXiv:quantph/0307205.
  13. A. Coladangelo, K. T. Goh, and V. Scarani, All pure bipartite entangled states can be self-tested, Nat. Commun. 8, 15485 (2017).
  14. A. Rai, M. Pivoluska, M. Plesch, S. Sasmal, M. Banik, and S. Ghosh, Device-independent bounds from Cabello's nonlocality argument, Phys. Rev. A 103, 062219 (2021).
  15. L. Wooltorton, P. Brown, and R. Colbeck, Tight analytic bound on the trade-off between device-independent randomness and nonlocality, Phys. Rev. Lett. 129, 150403 (2022).
  16. A. Rai, M. Pivoluska, S. Sasmal, M. Banik, S. Ghosh, and M. Plesch, Self-testing quantum states via nonmaximal violation in Hardy's test of nonlocality, Phys. Rev. A 105, 052227 (2022).
  17. C. Bamps and S. Pironio, Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing, Phys. Rev. A 91, 052111 (2015).
  18. M. McKague, Self-testing in parallel, New J. Phys. 18, 045013 (2016).
  19. X. Wu, J.-D. Bancal, M. McKague, and V. Scarani, Device-independent parallel self-testing of two singlets, Phys. Rev. A 93, 062121 (2016).
  20. X. Wu, Y. Cai, T. H. Yang, H. N. Le, J.-D. Bancal, and V. Scarani, Robust self-testing of the three-qubit w state, Phys. Rev. A 90, 042339 (2014).
  21. E. Panwar, P. Pandya, and M. Wieśniak, An elegant scheme of self-testing for multipartite Bell inequalities, npj Quantum Inf. 9, 71 (2023).
  22. R. K. Singh, S. Sasmal, and A. K. Pan, Self-testing of the m-partite Greenberger-Horne-Zeilinger state and observables using the Svetlichny inequality, Phys. Rev. A 112, 032404 (2025).
  23. S. Sarkar, D. Saha, J. Kaniewski, and R. Augusiak, Self-testing quantum systems of arbitrary local dimension with minimal number of measurements, npj Quantum Inf. 7, 151 (2021).
  24. M. McKague, Self-testing graph states, in Theory of Quantum Computation, Communication, and Cryptography: 6th Conference, TQC 2011, Revised Selected Papers (Lecture Notes in Computer Science, Volume 6745), edited by D. Bacon, M. Roetteler, and M. Martin-Delgado (Springer, Germany, 2014), pp. 104–120.
  25. J. Bowles, I. Šupić, D. Cavalcanti, and A. Acín, Device-independent entanglement certification of all entangled states, Phys. Rev. Lett. 121, 180503 (2018).
  26. A. K. Pan, Oblivious communication game, self-testing of projective and nonprojective measurements, and certification of randomness, Phys. Rev. A 104, 022212 (2021).
  27. J. Bowles, I. Šupić, D. Cavalcanti, and A. Acín, Self-testing of Pauli observables for device-independent entanglement certification, Phys. Rev. A 98, 042336 (2018).
  28. S. Wagner, J.-D. Bancal, N. Sangouard, and P. Sekatski, Device-independent characterization of quantum instruments, Quantum 4, 243 (2020).
  29. K. Mohan, A. Tavakoli, and N. Brunner, Sequential random access codes and self-testing of quantum measurement instruments, New J. Phys. 21, 083034 (2019).
  30. P. Sekatski, J.-D. Bancal, S. Wagner, and N. Sangouard, Certifying the building blocks of quantum computers from Bell's theorem, Phys. Rev. Lett. 121, 180505 (2018).
  31. P. Sekatski, J.-D. Bancal, M. Ioannou, M. Afzelius, and N. Brunner, Toward the device-independent certification of a quantum memory, Phys. Rev. Lett. 131, 170802 (2023).
  32. P. Roy and A. K. Pan, Device-independent self-testing of unsharp measurements, New J. Phys. 25, 013040 (2023).
  33. R. Paul, S. Sasmal, and A. K. Pan, Self-testing of multiple unsharpness parameters through sequential violations of a noncontextual inequality, Phys. Rev. A 110, 012444 (2024).
  34. E. S. Gómez, S. Gómez, P. González, G. Cañas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. Vértesi, and G. Lima, Device-independent certification of a nonprojective qubit measurement, Phys. Rev. Lett. 117, 260401 (2016).
  35. I. Šupić, J. Bowles, M.-O. Renou, A. Acín, and M. J. Hoban, Quantum networks self-test all entangled states, Nat. Phys. 19, 670 (2023).
  36. S. Munshi and A. K. Pan, Characterizing nonlocal correlations through various n-locality inequalities in a quantum network, Phys. Rev. A 105, 032216 (2022).
  37. S. Munshi and A. K. Pan, Self-testing of an unbounded number of mutually commuting local observables, Phys. Rev. A 108, 062607 (2023).
  38. M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acín, and M. Navascués, Quantum theory based on real numbers can be experimentally falsified, Nature (London) 600, 625 (2021).
  39. A. Tavakoli, J. Kaniewski, T. Vértesi, D. Rosset, and N. Brunner, Self-testing quantum states and measurements in the prepare-and-measure scenario, Phys. Rev. A 98, 062307 (2018).
  40. N. Miklin, J. J. Borkała, and M. Pawłowski, Semi-device-independent self-testing of unsharp measurements, Phys. Rev. Res. 2, 033014 (2020).
  41. I. Šupić, M. J. Hoban, L. D. Colomer, and A. Acín, Self-testing and certification using trusted quantum inputs, New J. Phys. 22, 073006 (2020).
  42. A. S. S., S. Mukherjee, and A. K. Pan, Robust certification of unsharp instruments through sequential quantum advantages in a prepare-measure communication game, Phys. Rev. A 107, 012411 (2023).
  43. R. K. Singh, S. Sasmal, S. Nautiyal, and A. K. Pan, Self-testing in a constrained prepare-measure scenario sans assuming quantum dimension, New J. Phys. 27, 114520 (2025).
  44. R. Paul, P. Roy, and A. K. Pan, Semi-device-independent self-testing of unitary operations, Phys. Rev. A 112, 062216 (2025).
  45. M. Smania, P. Mironowicz, M. Nawareg, M. Pawłowski, A. Cabello, and M. Bourennane, Experimental certification of an informationally complete quantum measurement in a device-independent protocol, Optica 7, 123 (2020).
  46. S. Gómez, A. Mattar, E. S. Gómez, D. Cavalcanti, O. J. Farías, A. Acín, and G. Lima, Experimental nonlocality-based randomness generation with nonprojective measurements, Phys. Rev. A 97, 040102(R) (2018).
  47. W.-H. Zhang, G. Chen, X.-X. Peng, X.-J. Ye, P. Yin, Y. Xiao, Z.-B. Hou, Z.-D. Cheng, Y.-C. Wu, J.-S. Xu, C.-F. Li, and G.-C. Guo, Experimentally robust self-testing for bipartite and tripartite entangled states, Phys. Rev. Lett. 121, 240402 (2018).
  48. M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y.-S. Zhang, Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement, npj Quantum Inf. 4, 63 (2018).
  49. X.-M. Hu, Y. Xie, A. S. Arora, M.-Z. Ai, K. Bharti, J. Zhang, W. Wu, P.-X. Chen, J.-M. Cui, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, J. Roland, A. Cabello, and L.-C. Kwek, Self-testing of a single quantum system from theory to experiment, npj Quantum Inf. 9, 103 (2023).
  50. I. Šupić and J. Bowles, Self-testing of quantum systems: A review, Quantum 4, 337 (2020).
  51. M. McKague, T. H. Yang, and V. Scarani, Robust self-testing of the singlet, J. Phys. A: Math. Theor. 45, 455304 (2012).
  52. J. Kaniewski, Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities, Phys. Rev. Lett. 117, 070402 (2016).
  53. I. Šupić, R. Augusiak, A. Salavrakos, and A. Acín, Self-testing protocols based on the chained Bell inequalities, New J. Phys. 18, 035013 (2016).
  54. S. Sarkar and R. Augusiak, Self-testing of multipartite Greenberger-Horne-Zeilinger states of arbitrary local dimension with arbitrary number of measurements per party, Phys. Rev. A 105, 032416 (2022).
  55. W.-H. Zhang, G. Chen, P. Yin, X.-X. Peng, X.-M. Hu, Z.-B. Hou, Z.-Y. Zhou, S. Yu, X.-J. Ye, Z.-Q. Zhou, X.-Y. Xu, J.-S. Tang, J.-S. Xu, Y.-J. Han, B.-H. Liu, C.-F. Li, and G.-C. Guo, Experimental demonstration of robust self-testing for bipartite entangled states, npj Quantum Inf. 5, 4 (2019).
  56. I. Agresti, B. Polacchi, D. Poderini, E. Polino, A. Suprano, I. Šupić, J. Bowles, G. Carvacho, D. Cavalcanti, and F. Sciarrino, Experimental robust self-testing of the state generated by a quantum network, PRX Quantum 2, 020346 (2021).
  57. C. Jordan, Essay on geometry in n dimensions, Bul. Soc. Math. France 3, 103 (1875).
  58. S. Wehner, Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities, Phys. Rev. A 73, 022110 (2006).
  59. Y. Xiao, X. H. Li, J. Wang, M. Li, and S.-M. Fei, Device-independent randomness based on a tight upper bound of the maximal quantum value of chained inequality, Phys. Rev. A 107, 052415 (2023).
  60. F. Baccari, R. Augusiak, I. Šupić, J. Tura, and A. Acín, Scalable Bell inequalities for qubit graph states and robust self-testing, Phys. Rev. Lett. 124, 020402 (2020).
  61. M. M. Wilde, Quantum Information Theory (Cambridge University Press, Cambridge, UK, 2013).
  62. R. Colbeck, Quantum and relativistic protocols for secure multi-party computation, arXiv:0911.3814.
  63. S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random numbers certified by Bell's theorem, Nature (London) 464, 1021 (2010).
  64. C. Dhara, G. Prettico, and A. Acín, Maximal quantum randomness in Bell tests, Phys. Rev. A 88, 052116 (2013).

Outline

Information

Phys. Rev. A 113, 042211– Published 14 April, 2026
  • Received 28 August 2025
  • Accepted 27 February 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation