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General concurrence percolation on quantum networks

Deep Nath* and Soumen Roy†

  • Department of Physical Sciences, Bose Institute, Kolkata, West Bengal 700091, India

  • *Contact author: deepnath441@gmail.com
  • †Contact author: soumen@jcbose.ac.in
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Phys. Rev. A 114, 012440 – Published 16 July, 2026

DOI: https://doi.org/10.1103/2sf7-srwz

Abstract

A quantum network is a network of entangled states which can be used to transmit quantum information. Nonmaximally entangled states are not really effective in establishing quantum communication across vast distances. Creating and maintaining a maximally entangled state over which quantum information can be transferred more effectively is difficult in experiments, as nonmaximally entangled states are typically produced therein. Therefore, we usually construct a network of nonmaximally entangled states and use the underlying network structure to establish maximal entanglement between distant nodes. Various protocols leveraging interesting aspects of quantum and statistical physics are designed to achieve long-range maximally entangled states. Here, we consider two-dimensional lattices as quantum networks where edges represent nonmaximally entangled pure states. We introduce a general protocol based on bond percolation—general concurrence percolation (GCP)—to create a network of maximally entangled states. The associated network obtained through GCP ought to enable the achievement of long-range quantum communication. Our observations indicate that, similar to other existing protocols, GCP falls within the percolation universality class, as determined through finite-size scaling analysis performed on two-dimensional lattices. We find that the associated percolation threshold value is smaller than that of the existing protocols. We introduce and analytically solve a minimal model that renders insight into the origin of the fundamentally lower percolation threshold achieved by GCP.

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References (36)

  1. H. Buhrman, R. Cleve, and A. Wigderson, Quantum vs. classical communication and computation, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1998), pp. 63–68.
  2. N. Gisin and R. Thew, Quantum communication, Nat. Photon. 1, 165 (2007).
  3. F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: A review, Rep. Prog. Phys. 82, 016001 (2019).
  4. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  5. W. K. Wootters, Quantum entanglement as a quantifiable resource, Philos. Trans. A Math. Phys. Eng. Sci. 356, 1717 (1998).
  6. R. Penrose, Quantum computation, entanglement and state reduction, Philos. Trans. A Math. Phys. Eng. Sci. 356, 1927 (1998).
  7. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993).
  8. J. Yin, Y. H. Li, S. K. Liao, M. Yang, Y. Cao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, S. L. Li, et al., Entanglement-based secure quantum cryptography over 1,120 kilometres, Nature (London) 582, 501 (2020).
  9. D. Watts, J. Bordes, J. Brown, A. Cherlin, R. Newton, J. Allison, M. Bashkanov, N. Efthimiou, and N. Zachariou, Photon quantum entanglement in the MeV regime and its application in PET imaging, Nat. Commun. 12, 2646 (2021).
  10. C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017).
  11. A. Ekert and R. Jozsa, Quantum algorithms: Entanglement–enhanced information processing, Philos. Trans. A Math. Phys. Eng. Sci. 356, 1769 (1998).
  12. C. Simon, Towards a global quantum network, Nat. Photon. 11, 678 (2017).
  13. S. Brito, A. Canabarro, R. Chaves, and D. Cavalcanti, Statistical properties of the quantum internet, Phys. Rev. Lett. 124, 210501 (2020).
  14. M. Schlosshauer, Quantum decoherence, Phys. Rep. 831, 1 (2019).
  15. J. I. de Vicente, Maximally entangled mixed states for a fixed spectrum do not always exist, Phys. Rev. Lett. 133, 050202 (2024).
  16. S. P. Neumann, S. K. Joshi, M. Fink, T. Scheidl, R. Blach, C. Scharlemann, S. Abouagaga, D. Bambery, E. Kerstel, M. Barthelemy, et al., Q3Sat: Quantum communications uplink to a 3U CubeSat—Feasibility & design, EPJ Quantum Technol. 5, 4 (2018).
  17. Z. Zuo, Y. Wang, Q. Liao, and Y. Guo, Overcoming the uplink limit of satellite-based quantum communication with deterministic quantum teleportation, Phys. Rev. A 104, 022615 (2021).
  18. S. Brito, A. Canabarro, D. Cavalcanti, and R. Chaves, Satellite-based photonic quantum networks are small-world, PRX Quantum 2, 010304 (2021).
  19. S. Perseguers, G. Lapeyre, D. Cavalcanti, M. Lewenstein, and A. Acín, Distribution of entanglement in large-scale quantum networks, Rep. Prog. Phys. 76, 096001 (2013).
  20. M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors” Bell experiment via entanglement swapping, Phys. Rev. Lett. 71, 4287 (1993).
  21. S. Bose, V. Vedral, and P. L. Knight, Purification via entanglement swapping and conserved entanglement, Phys. Rev. A 60, 194 (1999).
  22. E. Shchukin and P. van Loock, Optimal entanglement swapping in quantum repeaters, Phys. Rev. Lett. 128, 150502 (2022).
  23. A. Acín, J. I. Cirac, and M. Lewenstein, Entanglement percolation in quantum networks, Nat. Phys. 3, 256 (2007).
  24. X. Meng, J. Gao, and S. Havlin, Concurrence percolation in quantum networks, Phys. Rev. Lett. 126, 170501 (2021).
  25. O. Malik, X. Meng, S. Havlin, G. Korniss, B. K. Szymanski, and J. Gao, Concurrence percolation threshold of large-scale quantum networks, Commun. Phys. 5, 193 (2022).
  26. S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr, Entanglement distribution in pure-state quantum networks, Phys. Rev. A 77, 022308 (2008).
  27. D. Stauffer and A. Aharony, Introduction to Percolation Theory (CRC Press, Boca Raton, FL, 1994).
  28. A. Malthe-Sørenssen, Percolation Theory Using Python (Springer, Cham, 2024).
  29. M. C. Gimenez, L. Reinaudi, and P. M. Centres, Percolation threshold and critical exponent analysis in equilibrium systems on simple cubic and bcc lattices, Phys. A 668, 130562 (2025).
  30. E. W. Dijkstra, A note on two problems in connexion with graphs, Numer. Math. 1, 269 (1959).
  31. Code for general concurrence percolation on quantum networks, https://github.com/deepnath441/GCP.
  32. K. Binder, Monte Carlo Simulation in Statistical Physics (Springer-Verlag, New York, 1992).
  33. X. Hu, G. Dong, K. Christensen, H. Sun, J. Fan, Z. Tian, J. Gao, S. Havlin, R. Lambiotte, and X. Meng, Unveiling the importance of nonshortest paths in quantum networks, Sci. Adv. 11, eadt2404 (2025).
  34. M. Cuquet and J. Calsamiglia, Entanglement percolation in quantum complex networks, Phys. Rev. Lett. 103, 240503 (2009).
  35. S. Perseguers, D. Cavalcanti, G. Lapeyre, Jr., M. Lewenstein, and A. Acín, Multipartite entanglement percolation, Phys. Rev. A 81, 032327 (2010).
  36. M. E. J. Newman and R. M. Ziff, Fast Monte Carlo algorithm for site or bond percolation, Phys. Rev. E 64, 016706 (2001).

Outline

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Phys. Rev. A 114, 012440– Published 16 July, 2026
  • Received 28 December 2025
  • Revised 2 May 2026
  • Accepted 1 July 2026
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