- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Unbounded-input explicit Bell inequalities for general quantum networks

Yao Xiao1,2,3, Fenzhuo Guo1,2,3,*, Haifeng Dong4, Sujuan Qin3, and Fei Gao3

  • *Contact author: gfenzhuo@bupt.edu.cn
PDF

Phys. Rev. Research 8, 013121 – Published 3 February, 2026

DOI: https://doi.org/10.1103/w2yy-mhp3

Abstract

Quantum nonlocality in networks featuring multiple independent sources underpins large-scale quantum communication and poses fundamental challenges for its characterization. In this work, we construct a family of explicit nonlinear Bell inequalities to verify the nonlocality across the general multi-input quantum networks. The construction of these inequalities relies on the number of leaf nodes, a network parameter that can be identified by a linear-time algorithm. Our approach establishes a structural connection between bipartite full-correlation Bell inequalities and network Bell inequalities, enabling the analytical derivation of optimal quantum violations and the conditions under which they occur. We further quantify the upper bound on maximal violations achievable by arbitrary two-qubit mixed states in such networks, under separable measurements, and evaluate the noise robustness of the proposed inequalities via the visibilities of noisy states. Finally, we demonstrate that these inequalities can, in a device-independent manner, distinguish between network topologies of equal size that differ in the number of leaf nodes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (42)

  1. J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  2. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  3. 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).
  4. 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, et al., Random numbers certified by Bell's theorem, Nature (London) 464, 1021 (2010).
  5. H. Buhrman, R. Cleve, S. Massar, and R. de Wolf, Nonlocality and communication complexity, Rev. Mod. Phys. 82, 665 (2010).
  6. D. Rosset, J.-D. Bancal, and N. Gisin, Classifying 50 years of Bell inequalities, J. Phys. A: Math. Theor. 47, 424022 (2014).
  7. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed. (Cambridge University Press, Cambridge, England, 2004).
  8. R. Cleve and H. Buhrman, Substituting quantum entanglement for communication, Phys. Rev. A 56, 1201 (1997).
  9. A. Tavakoli, A. Pozas-Kerstjens, M-X. Luo, and M-O. Renou, Bell nonlocality in networks, Rep. Prog. Phys. 85, 056001 (2022).
  10. C. Branciard, N. Gisin, and S. Pironio, Characterizing the nonlocal correlations created via entanglement swapping, Phys. Rev. Lett. 104, 170401 (2010).
  11. C. Branciard, D. Rosset, N. Gisin, and S. Pironio, Bilocal versus nonbilocal correlations in entanglement-swapping experiments, Phys. Rev. A 85, 032119 (2012).
  12. L.-Y. Hsu, Nonlocal correlations in quantum networks distributed with different entangled states, New J. Phys. 26, 033026 (2024).
  13. A. Tavakoli, P. Skrzypczyk, D. Cavalcanti, and A. Acín, Nonlocal correlations in the star-network configuration, Phys. Rev. A 90, 062109 (2014).
  14. A. Tavakoli, M.-O. Renou, N. Gisin, and N. Brunner, Correlations in star networks: From Bell inequalities to network inequalities, New J. Phys. 19, 073003 (2017).
  15. S. Munshi, R. Kumar, and A. K. Pan, Generalized n-locality inequalities in a star-network configuration and their optimal quantum violations, Phys. Rev. A 104, 042217 (2021).
  16. S. Munshi and A. K. Pan, Characterizing nonlocal correlations through various n-locality inequalities in a quantum network, Phys. Rev. A 105, 032216 (2022).
  17. K. Mukherjee, B. Paul, and D. Sarkar, Correlations in n-local scenario, Quantum Inf. Process. 14, 2025 (2015).
  18. R. Kumar and A. K. Pan, Generalized n-locality inequalitiesin linear-chain network for arbitrary inputs scenario andtheir quantum violations, Ann. Phys. 534, 2200182 (2022).
  19. L. Yang, X. Qi, and J. Hou, Nonlocal correlations in the tree-tensor-network configuration, Phys. Rev. A 104, 042405 (2021).
  20. L. Yang, X. Qi, and J. Hou, Quantum nonlocality in any forked tree-shaped network, Entropy 24, 691 (2022).
  21. L. Yang, X. Qi, and J. Hou, Generalized n-locality correlations in tree tensor network configuration, Eur. Phys. J. Plus 138, 772 (2023).
  22. R. Chaves, Polynomial Bell inequalities, Phys. Rev. Lett. 116, 010402 (2016).
  23. C. M. Lee and R. W. Spekkens, Causal inference via algebraic geometry: Feasibility tests for functional causal structures with two binary observed variables, J. Causal Infer. 5, 20160013 (2017).
  24. D. Rosset, C. Branciard, T. J. Barnea, G. Pütz, N. Brunner, and N. Gisin, Nonlinear Bell inequalities tailored for quantum networks, Phys. Rev. Lett. 116, 010403 (2016).
  25. A. Tavakoli, Bell-type inequalities for arbitrary noncyclic networks, Phys. Rev. A 93, 030101(R) (2016).
  26. M-X. Luo, Computationally efficient nonlinear Bell inequalities for quantum networks, Phys. Rev. Lett. 120, 140402 (2018).
  27. S-Y. Yang, K. He, J-C. Hou, Z-H. Ma, S-M. Fei, and M-X. Luo, Witnessing network topologies using quantum nonlocality, Phys. Rev. A 110, 032437 (2024).
  28. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms (MIT Press, Cambridge, MA, 2009).
  29. B. Doolittle and E. Chitambar, Maximal qubit violations of n-locality in star and chain networks, Phys. Rev. A 108, 042409 (2023).
  30. N. Gisin, Q. Mei, A. Tavakoli, M. O. Renou, and N. Brunner, All entangled pure quantum states violate the bilocality inequality, Phys. Rev. A 96, 020304(R) (2017).
  31. F. Andreoli1, G. Carvacho, L. Santodonato, R. Chaves, and F. Sciarrino, Maximal qubit violation of n-locality inequalities in a star-shaped quantum network, New J. Phys. 19, 113020 (2017).
  32. A. Kundu, M. K. Molla, I. Chattopadhyay, and D. Sarkar, Maximal qubit violation of n-local inequalities in a quantum network, Phys. Rev. A 102, 052222 (2020).
  33. D. Poderini, I. Agresti, G. Marchese, E. Polino, T. Giordani, A. Suprano, M. Valeri, G. Milani, N. Spagnolo, G. Carvacho, et al., Experimental violation of n-locality in a star quantum network, Nat. Commun. 11, 2467 (2020).
  34. I. šupić, R. Augusiak, A. Salavrakos, and A Acín, Self-testing protocols based on the chained Bell inequalities, New J. Phys. 18, 035013 (2016).
  35. N. Gisin, Inquantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony (Springer, Netherlands, Dordrecht, 2009), p. 125.
  36. S. Munshi and A. K. Pan, Device-independent full network nonlocality for arbitrary-party and unbounded-input scenario, Phys. Rev. Lett. 134, 210203 (2025).
  37. 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).
  38. S. L. Braunstein and C. M. Caves, Wringing out better Bell inequalities, Ann. Phys. 202, 22 (1990).
  39. S. Ghorai and A. K. Pan, Optimal quantum preparation contextuality in an n-bit parity-oblivious multiplexing task, Phys. Rev. A 98, 032110 (2018).
  40. R. Horodecki, P. Horodecki, and M. Horodecki, Violating Bell inequality by mixed spin-12 states: Necessary and sufficient condition, Phys. Lett. A 200, 340 (1995).
  41. A. Acín, S. Pironio, T. Vértesi, and P. Wittek, Optimal randomness certification from one entangled bit, Phys. Rev. A 93, 040102(R) (2016).
  42. A. K. Pan and S. S. Mahato, Device-independent certification of the Hilbert-space dimension using a family of Bell expressions, Phys. Rev. A 102, 052221 (2020).

Outline

Information

Phys. Rev. Research 8, 013121– Published 3 February, 2026
  • Received 9 September 2025
  • Accepted 13 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation