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Composable finite-size security of high-dimensional quantum-key-distribution protocols

Florian Kanitschar1,2,* and Marcus Huber1,3

  • *Contact author: florian.kanitschar@outlook.com
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Phys. Rev. Applied 24, 054028 – Published 10 November, 2025

DOI: https://doi.org/10.1103/v51y-vkfr

Abstract

Practical implementations of quantum key distribution (QKD) extending beyond urban areas commonly use satellite links; however, the transmission of quantum states through the Earth’s atmosphere is highly susceptible to noise, restricting its application primarily to nighttime. High-dimensional (HD) QKD offers a promising solution to this limitation by employing high-dimensionally entangled quantum states. Although experimental platforms for HD QKD exist, previous security analyses have been limited to the asymptotic regime and have either relied on impractical measurements or employed computationally demanding convex optimization tasks, restricting the security analysis to low dimensions. In this work, we bridge this gap by presenting a composable finite-size security proof against both collective and coherent attacks for a general HD QKD protocol that uses only experimentally accessible measurements. In addition to the conventional yet impractical “one-shot” key rates, we provide a practical variable-length security argument that yields significantly higher expected key rates. This approach is particularly crucial under rapidly changing and turbulent atmospheric conditions, as encountered in free-space and satellite-based QKD platforms.

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References (57)

  1. C. H. Bennett, and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, India, 1984), p. 175.
  2. A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67, 661 (1991).
  3. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys. 81, 1301 (2009).
  4. S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, Advances in quantum cryptography, Adv. Opt. Photonics 12, 1012 (2020).
  5. V. C. Usenko, A. Acín, R. Alléaume, U. L. Andersen, E. Diamanti, T. Gehring, A. A. E. Hajomer, F. Kanitschar, C. Pacher, S. Pirandola, and V. Pruneri, Continuous-variable quantum communication, ArXiv:2501.12801.
  6. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, Long-distance quantum key distribution in optical fibre, New J. Phys. 8, 193 (2006).
  7. M. Peev et al., The SECOQC quantum key distribution network in Vienna, New J. Phys. 11, 075001 (2009).
  8. D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, Long-distance decoy-state quantum key distribution in optical fiber, Phys. Rev. Lett. 98, 010503 (2007).
  9. C.-Z. Peng, J. Zhang, D. Yang, W.-B. Gao, H.-X. Ma, H. Yin, H.-P. Zeng, T. Yang, X.-B. Wang, and J.-W. Pan, Experimental long-distance decoy-state quantum key distribution based on polarization encoding, Phys. Rev. Lett. 98, 010505 (2007).
  10. A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, Secure quantum key distribution over 421 km of optical fiber, Phys. Rev. Lett. 121, 190502 (2018).
  11. T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, Experimental demonstration of free-space decoy-state quantum key distribution over 144 km, Phys. Rev. Lett. 98, 010504 (2007).
  12. A. Kržič et al., Towards metropolitan free-space quantum networks, Npj Quantum Inf. 9, 1 (2023).
  13. R. Haber, D. Garbe, S. Busch, W. Rosenfeld, and K. Schilling, Qube - A CubeSat for Quantum Key Distribution Experiments, https://www.semanticscholar.org/paper/Qube-A-CubeSat-for-Quantum-Key-Distribution-Haber-Garbe/d3da9d85ef68c90ea8a1a1be96d047b7c61bfdab (2018).
  14. C.-Y. Lu, Y. Cao, C.-Z. Peng, and J.-W. Pan, Micius quantum experiments in space, Rev. Mod. Phys. 94, 035001 (2022).
  15. T. Jennewein, C. Simon, A. Fougeres, F. Babin, F. K. Asadi, K. B. Kuntz, M. Maisonneuve, B. Moffat, K. Mohammadi, and D. Panneton, QEYSSat 2.0 – White Paper on Satellite-based Quantum Communication Missions in Canada, Arxiv:2306.02481v2.
  16. J. S. Sidhu, T. Brougham, D. McArthur, R. G. Pousa, and D. K. L. Oi, Finite key performance of satellite quantum key distribution under practical constraints, Commun. Phys. 6, 1 (2023).
  17. D. Orsucci, P. Kleinpaß, J. Meister, I. De Marco, S. Häusler, T. Strang, N. Walenta, and F. Moll, Assessment of practical satellite quantum key distribution architectures for current and near-future missions, ArXiv:2404.05668.
  18. P. G. Kwiat, Hyper-entangled states, J. Mod. Opt. 44, 2173 (1997).
  19. J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, Generation of hyperentangled photon pairs, Phys. Rev. Lett. 95, 260501 (2005).
  20. A. Martin, T. Guerreiro, A. Tiranov, S. Designolle, F. Fröwis, N. Brunner, M. Huber, and N. Gisin, Quantifying photonic high-dimensional entanglement, Phys. Rev. Lett. 118, 110501 (2017).
  21. S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, Overcoming noise in entanglement distribution, Phys. Rev. X 9, 041042 (2019).
  22. L. Sheridan, and V. Scarani, Security proof for quantum key distribution using qudit systems, Phys. Rev. A 82, 030301(R) (2010).
  23. A. Ferenczi, and N. Lütkenhaus, Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning, Phys. Rev. A 85, 052310 (2012).
  24. M. Doda, M. Huber, G. Murta, M. Pivoluska, M. Plesch, and C. Vlachou, Quantum key distribution overcoming extreme noise: Simultaneous subspace coding using high-dimensional entanglement, Phys. Rev. Appl. 15, 034003 (2021).
  25. N. Wyderka, G. Chesi, H. Kampermann, C. Macchiavello, and D. Bruß, High-dimensional quantum key distribution rates for multiple measurement bases, ArXiv:2501.05890.
  26. T. N. Thomas, and W. O. Krawec, New key rate bound for high-dimensional BB84 with multiple basis measurements, ArXiv:2504.11315.
  27. F. Kanitschar, A. Bergmayr-Mann, M. Pivoluska, and M. Huber, Harnessing high-dimensional temporal entanglement using limited interferometric setups, Phys. Rev. Appl. 22, 054054 (2024).
  28. P. J. Coles, E. M. Metodiev, and N. Lütkenhaus, Numerical approach for unstructured quantum key distribution, Nat. Commun. 7, 11712 (2016).
  29. A. Winick, N. Lütkenhaus, and P. J. Coles, Reliable numerical key rates for quantum key distribution, Quantum 2, 77 (2018).
  30. M. Araújo, M. Huber, M. Navascués, M. Pivoluska, and A. Tavakoli, Quantum key distribution rates from semidefinite programming, Quantum 7, 1019 (2023).
  31. F. Kanitschar, and M. Huber, Practical framework for analyzing high-dimensional quantum key distribution setups, Phys. Rev. Lett. 135, 010802 (2025).
  32. J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, Measurements in two bases are sufficient for certifying high-dimensional entanglement, Nat. Phys. 14, 1032 (2018).
  33. X.-M. Hu, W.-B. Xing, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, P. Erker, and M. Huber, Efficient generation of high-dimensional entanglement through multipath down-conversion, Phys. Rev. Lett. 125, 090503 (2020).
  34. J. Schneeloch, C. C. Tison, M. L. Fanto, P. M. Alsing, and G. A. Howland, Quantifying entanglement in a 68-billion-dimensional quantum state space, Nat. Commun. 10, 2785 (2019).
  35. M. C. Ponce, A. L. M. Muniz, M. Huber, and F. Steinlechner, Unlocking the frequency domain for high-dimensional quantum information processing, ArXiv:2206.00969.
  36. A. S. Maxwell, L. B. Madsen, and M. Lewenstein, Entanglement of orbital angular momentum in non-sequential double ionization, Nat. Commun. 13, 4706 (2022).
  37. M. Christandl, R. König, and R. Renner, Postselection technique for quantum channels with applications to quantum cryptography, Phys. Rev. Lett. 102, 020504 (2009).
  38. S. Nahar, D. Tupkary, Y. Zhao, N. Lütkenhaus, and E. Y. Z. Tan, Postselection technique for optical quantum key distribution with improved de Finetti reductions, ArXiv:2403.11851.
  39. D. Tupkary, E. Y.-Z. Tan, and N. Lütkenhaus, Security proof for variable-length quantum key distribution, Phys. Rev. Res. 6, 023002 (2024).
  40. I. George, J. Lin, and N. Lütkenhaus, Numerical calculations of the finite key rate for general quantum key distribution protocols, Phys. Rev. Res. 3, 013274 (2021).
  41. F. Kanitschar, I. George, J. Lin, T. Upadhyaya, and N. Lütkenhaus, Finite-size security for discrete-modulated continuous-variable quantum key distribution protocols, PRX Quantum 4, 040306 (2023).
  42. R. Renner, Security of Quantum Key Distribution, Ph.D. thesis, ETH Zürich, Zürich, Switzerland, 2005, ArXiv:quant-ph/0512258.
  43. C. Portmann, and R. Renner, Security in quantum cryptography, Rev. Mod. Phys. 94, 025008 (2022).
  44. M. Tomamichel, and A. Leverrier, A largely self-contained and complete security proof for quantum key distribution, Quantum 1, 14 (2017).
  45. M. Tomamichel, A framework for non-asymptotic quantum information theory, ArXiv:1203.2142.
  46. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc. 58, 13 (1963).
  47. A. Tiranov, S. Designolle, E. Z. Cruzeiro, J. Lavoie, N. Brunner, M. Afzelius, M. Huber, and N. Gisin, Quantification of multidimensional entanglement stored in a crystal, Phys. Rev. A 96, 040303(R) (2017).
  48. J. Lin, T. Upadhyaya, and N. Lütkenhaus, Asymptotic security analysis of discrete-modulated continuous-variable quantum key distribution, Phys. Rev. X 9, 041064 (2019).
  49. T. Upadhyaya, T. van Himbeeck, J. Lin, and N. Lütkenhaus, Dimension reduction in quantum key distribution for continuous- and discrete-variable protocols, PRX Quantum 2, 020325 (2021).
  50. F. Kanitschar, Composable-Finite-Size-Security-of-High-Dimensional-Quantum-Key-Distribution-Protocols, 2025, https://github.com/fkan95/Composable-Finite-Size-Security-of-High-Dimensional-Quantum-Key-Distribution-Protocols
  51. L. Bulla, M. Pivoluska, K. Hjorth, O. Kohout, J. Lang, S. Ecker, S. P. Neumann, J. Bittermann, R. Kindler, M. Huber, M. Bohmann, and R. Ursin, Nonlocal temporal interferometry for highly resilient free-space quantum communication, Phys. Rev. X 13, 021001 (2023).
  52. N. T. Islam, C. C. W. Lim, C. Cahall, J. Kim, and D. J. Gauthier, Provably secure and high-rate quantum key distribution with time-bin qudits, Sci. Adv. 3, e1701491 (2017).
  53. L. Bulla, K. Hjorth, O. Kohout, J. Lang, S. Ecker, S. P. Neumann, J. Bittermann, R. Kindler, M. Huber, M. Bohmann, R. Ursin, and M. Pivoluska, Distribution of genuine high-dimensional entanglement over 10.2 km of noisy metropolitan atmosphere, Phys. Rev. A 107, L050402 (2023).
  54. K. Sulimany, G. Pelc, R. Dudkiewicz, S. Korenblit, H. S. Eisenberg, Y. Bromberg, and M. Ben-Or, High-dimensional coherent one-way quantum key distribution, Npj Quantum Inf. 11, 16 (2025).
  55. F. Bouchard, D. England, P. J. Bustard, K. L. Fenwick, E. Karimi, K. Heshami, and B. Sussman, Achieving ultimate noise tolerance in quantum communication, Phys. Rev. Appl. 15, 024027 (2021).
  56. S.-K. Liao et al., Long-distance free-space quantum key distribution in daylight towards inter-satellite communication, Nat. Photonics 11, 509 (2017).
  57. R. Bedington, J. M. Arrazola, and A. Ling, Progress in satellite quantum key distribution, npj Quantum Inf. 3, 30 (2017).

Outline

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Phys. Rev. Applied 24, 054028– Published 10 November, 2025
  • Received 8 February 2025
  • Revised 2 September 2025
  • Accepted 17 October 2025
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