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

Physical Review A

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
  • Access by Xinjiang University

Continuous-variable quantum key distribution with composable security and tight error-correction bound towards constrained-device implementations

Panagiotis Papanastasiou1,3, Carlo Ottaviani2,3, Stefano Pirandola2, Poonam Yadav2, and Marco Lucamarini1,3

PDF

Phys. Rev. A 114, 022611 – Published 11 August, 2026

DOI: https://doi.org/10.1103/2s9n-s2br

Abstract

Constrained devices, such as smart sensors, wearable devices, and Internet of Things nodes, are increasingly prevalent in society and rely on secure communications to function properly. These devices often operate autonomously, exchanging sensitive data or commands over short distances, such as within a room, house, or warehouse. In this context, continuous-variable quantum key distribution (CV-QKD) offers the highest secure key rate and the greatest versatility for integration into existing infrastructure. A key challenge in this setting, where devices have limited storage and processing capacity, is obtaining a realistic and tight estimate of the CV-QKD secure key rate within a composable security framework, with error correction (EC) consuming most of the storage and computational power. To address this, we focus on low-density parity-check codes with nonbinary alphabets, which optimize mutual information and are particularly suited for short-distance communications. We develop a security framework to derive finite-size secret keys near the optimal EC leakage limit and model the related memory requirements for the encoding process in one-way error correction. This analysis facilitates the practical deployment of CV-QKD, particularly in constrained devices with limited storage and computational resources.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (85)

  1. C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, Theor. Comput. Sci. 560, 7 (2014).
  2. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 145 (2002).
  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. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, Advances in quantum cryptography, Adv. Opt. Photon. 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, Rev. Mod. Phys. 98, 015003 (2026).
  6. Y. K. Liu and D. Moody, Post-quantum cryptography and the quantum future of cybersecurity, Phys. Rev. Appl. 21, 040501 (2024).
  7. C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  8. A. Leverrier, Security of continuous-variable quantum key distribution via a Gaussian de Finetti reduction, Phys. Rev. Lett. 118, 200501 (2017).
  9. S. Pirandola and P. Papanastasiou, Improved composable key rates for CV-QKD, Phys. Rev. Res. 6, 023321 (2024).
  10. Y. Zhang, Z. Li, Z. Chen, C. Weedbrook, Y. Zhao, X. Wang, Y. Huang, C. Xu, X. Zhang, Z. Wang, M. Li, X. Zhang, Z. Zheng, B. Chu, X. Gao, N. Meng, W. Cai, Z. Wang, G. Wang, S. Yu, et al., Continuous-variable QKD over 50 km commercial fiber, Quantum Sci. Technol. 4, 035006 (2019).
  11. T. Wang, P. Huang, Y. Zhou, W. Liu, H. Ma, S. Wang, and G. Zeng, High key rate continuous-variable quantum key distirbution with a real local oscillator, Opt. Express 26, 2794 (2018).
  12. S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, Fundamental limits of repeaterless quantum communications, Nat. Commun. 8, 15043 (2017).
  13. F. Grosshans and P. Grangier, Continuous variable quantum cryptography using coherent states, Phys. Rev. Lett. 88, 057902 (2002).
  14. C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, Quantum cryptography without switching, Phys. Rev. Lett. 93, 170504 (2004).
  15. F. Grosshans and P. Grangier, Reverse reconciliation protocols for quantum cryptography with continuous variables, arXiv:quant-ph/0204127.
  16. R. Filip, Continuous-variable quantum key distribution with noisy coherent states, Phys. Rev. A 77, 022310 (2008).
  17. S. Pirandola, S. Mancini, S. Lloyd, and S. L. Braunstein, Continuous variable quantum cryptography using two-way quantum communication, Nat. Phys. 4, 726 (2008).
  18. C. Weedbrook, C. Ottaviani, and S. Pirandola, Two-way quantum cryptography at different wavelengths, Phys. Rev. A 89, 012309 (2014).
  19. P. Papanastasiou, C. Ottaviani, and S. Pirandola, Gaussian one-way thermal quantum cryptography with finite-size effects, Phys. Rev. A 98, 032314 (2018).
  20. C. Ottaviani, M. J. Woolley, M. Erementchouk, J. F. Federici, P. Mazumder, S. Pirandola, and C. Weedbrook, Terahertz quantum cryptography, IEEE J. Select. Areas Commun. 38, 483 (2020).
  21. S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, High-rate measurement-device-independent quantum cryptography, Nat. Photon. 9, 397 (2015).
  22. Z. Li, Y.-C. Zhang, F. Xu, X. Peng, and H. Guo, Continuous-variable measurement-device-independent quantum key distribution, Phys. Rev. A 89, 052301 (2014).
  23. C. Ottaviani, G. Spedalieri, S. L. Braunstein, and S. Pirandola, Continuous-variable quantum cryptography with an untrusted relay: Detailed security analysis of the symmetric configuration, Phys. Rev. A 91, 022320 (2015).
  24. P. Papanastasiou, C. Ottaviani, and S. Pirandola, Finite-size analysis of measurement-device-independent quantum cryptography with continuous variables, Phys. Rev. A 96, 042332 (2017).
  25. X. Zhang, Y.-C. Zhang, Y. Zhao, X. Wang, S. Yu, and H. Guo, Finite-size analysis of continuous-variable measurement-device-independent quantum key distribution, Phys. Rev. A 96, 042334 (2017).
  26. C. Lupo, C. Ottaviani, P. Papanastasiou, and S. Pirandola, Parameter estimation with almost no public communication for continuous-variable quantum key distribution, Phys. Rev. Lett. 120, 220505 (2018).
  27. C. Lupo, C. Ottaviani, P. Papanastasiou, and S. Pirandola, Continuous-variable measurement-device-independent quantum key distribution: Composable security against coherent attacks, Phys. Rev. A 97, 052327 (2018).
  28. P. Papanastasiou, A. G. Mountogiannakis, and S. Pirandola, Composable security of CV-MDI-QKD with secret key rate and data processing, Sci. Rep. 13, 11636 (2023).
  29. A. I. Fletcher, C. Harney, M. Ghalaii, P. Papanastasiou, A. Mountogiannakis, G. Spedalieri, A. A. E. Hajomer, T. Gehring, and S. Pirandola, An overview of CV-MDI-QKD, Rep. Prog. Phys. 88, 084001 (2025).
  30. C. Ottaviani, C. Lupo, R. Laurenza, and S. Pirandola, Modular network for high-rate quantum conferencing, Commun. Phys. 2, 118 (2019). See, also, High-rate secure quantum conferencing, arXiv:1709.06988v2.
  31. P. Papanastasiou, C. Weedbrook, and S. Pirandola, Continuous-variable quantum key distribution in fast fading channels, Phys. Rev. A 97, 032311 (2018).
  32. M. Ghalaii, P. Papanastasiou, and S. Pirandola, Composable end-to-end security of Gaussian quantum networks with untrusted relays, npj Quantum Inf. 8, 105 (2022).
  33. T. Symul, D. J. Alton, S. M. Assad, A. M. Lance, C. Weedbrook, T. C. Ralph, and P. K. Lam, Experimental demonstration of post-selection-based continuous- variable quantum key distribution in the presence of Gaussian noise, Phys. Rev. A 76, 030303(R) (2007).
  34. K. N. Wilkinson, P. Papanastasiou, C. Ottaviani, T. Gehring, and S. Pirandola, Long-distance continuous-variable measurement-device-independent quantum key distribution with postselection, Phys. Rev. Res. 2, 033424 (2020).
  35. N. Hosseinidehaj, A. M. Lance, T. Symul, N. Walk, and T. C. Ralph, Finite-size effects in continuous-variable quantum key distribution with Gaussian postselection, Phys. Rev. A 101, 052335 (2020).
  36. A. Leverrier and P. Grangier, Continuous-variable quantum-key-distribution protocols with a non- Gaussian modulation, Phys. Rev. A 83, 042312 (2011).
  37. P. Papanastasiou, C. Lupo, C. Weedbrook, and S. Pirandola, Quantum key distribution with phase-encoded coherent states: Asymptotic security analysis in thermal-loss channels, Phys. Rev. A 98, 012340 (2018).
  38. S. Ghorai, P. Grangier, E. Diamanti, and A. Leverrier, Asymptotic security of continuous-variable quantum key distribution with a discrete modulation, Phys. Rev. X 9, 021059 (2019).
  39. P. Papanastasiou and S. Pirandola, Continuous-variable quantum cryptography with discrete alphabets: Composable security under collective Gaussian attacks, Phys. Rev. Res. 3, 013047 (2021).
  40. S. Bäuml, C. Pascual-García, V. Wright, O. Fawzi, and A. Acín, Security of discrete-modulated continuous-variable quantum key distribution, Quantum 8, 1418 (2024).
  41. A. A. E. Hajomer, F. Kanitschar, N. Jain, M. Hentschel, R. Zhang, N. Lütkenhaus, U. L. Andersen, C. Pacher, and T. Gehring, Experimental composable key distribution using discrete-modulated continuous variable quantum cryptography, Light Sci. Appl. 14, 255 (2025).
  42. A. A. E. Hajomer, A. Bomhals, C. Bruynsteen, A. Sidhique, O. Kovalenko, I. Derkach, V. C. Usenko, U. L. Andersen, X. Yin, and T. Gehring, Chip-based 16 GBaud continuous-variable quantum key distribution, Optica 13, 1035 (2026).
  43. A. A. E. Hajomer, C. Bruynsteen, I. Derkach, N. Jain, A. Bomhals, S. Bastiaens, U. L. Andersen, X. Yin, and T. Gehring, Continuous-variable quantum key distribution at 10 GBaud using an integrated photonic-electronic receiver, Optica 11, 1197 (2024).
  44. J. Aldama, S. Sarmiento, L. Trigo Vidarte, S. Etcheverry, I. López Grande, L. Castelvero, A. Hinojosa, T. Beckerwerth, Y. Piétri, A. Rhouni, E. Diamanti, and V. Pruneri, Integrated InP-based transmitter for continuous-variable quantum key distribution, Opt. Express 33, 8139 (2025).
  45. Y. Piétri, L. T. Vidarte, M. Schiavon, L. Vivien, P. Grangier, A. Rhouni, and E. Diamanti, Experimental demonstration of continuous-variable quantum key distribution with a silicon photonics integrated receiver, Opt. Quantum 2, 428 (2024).
  46. L. Li, T. Wang, X. Li, P. Huang, Y. Guo, L. Lu, L. Zhou, and G. Zeng, Continuous-variable quantum key distribution with on-chip light sources, Photon. Res. 11, 504 (2023).
  47. J. W. Silverstone, D. Bonneau, J. L. O'Brien, and M. G. Thompson, Silicon quantum photonics, IEEE J. Sel. Top. Quant. Electron. 22, 390 (2016).
  48. R. Roman, J. Zhou, and J. López, On the features and challenges of security and privacy in distributed Internet of things, Comput. Networks 57, 2266 (2013).
  49. N. Neshenko, E. Bou-Harb, J. Crichigno, G. Kaddoum, and N. Ghani, Demystifying IoT security: An exhaustive survey on IoT vulnerabilities and a first empirical look on Internet-scale IoT exploitations, IEEE Commun. Surv. Tutor. 21, 2702 (2019).
  50. G. Fitzgibbon and C. Ottaviani, Constrained device performance benchmarking with the implementation of post-quantum cryptography, Cryptography 8, 21 (2024).
  51. D. Joseph, R. Misoczki, M. Manzano, J. Tricot, F. D. Pinuaga, O. Lacombe, S. Leichenauer, J. Hidary, P. Venables, and R. Hansen, Transitioning organizations to post-quantum cryptography, Nature (London) 605, 237 (2022).
  52. M. Mosca, Cybersecurity in an era with quantum computers: Will we be ready? IEEE Secur. Priv. 16, 38 (2018).
  53. L. Garms, T. K. Paraïso, N. Hanley, A. Khalid, C. Rdafferty, J. Grant, J. Newman, A. J. Shields, C. Cid, and M. O'Neill, Experimental integration of quantum key distribution and post-quantum cryptography in a hybrid quantum-safe cryptosystem, Adv. Quantum Technol. 7, 2300304 (2024).
  54. A. K. Fedorov, Deploying hybrid quantum-secured infrastructure for applications: When quantum and post-quantum can work together, Front. Quantum Sci. Technol. 2, 1164428 (2023).
  55. J. S. Buruaga, A. Bugler, J. P. Brito, V. Martin, and C. Striecks, Versatile quantum-safe hybrid key exchange and its application to MACsec, EPJ Quantum Technol. 12, 84 (2025).
  56. X. Wang, Y. Zhang, S. Yu, and H. Guo, High speed error correction for continuous-variable quantum key distribution with multi-edge type LDPC code, Sci. Rep. 8, 10543 (2018).
  57. M. Milicevic, C. Feng, L. M. Zhang, and P. G. Gulak, Quasi-cyclic multi-edge LDPC codes for long-distance quantum cryptography, npj Quantum Inf. 4, 21 (2017).
  58. P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, Experimental demonstration of long-distance continuous-variable quantum key distribution, Nat. Photon. 7, 378 (2013).
  59. D. Huang, P. Huang, D. Lin, and G. Zeng, Long-distance continuous-variable quantum key distribution by controlling excess noise, Sci. Rep. 6, 19201 (2016).
  60. C. Pacher, J. Martinez-Mateo, J. Duhme, T. Gehring, and F. Furrer, Information reconciliation for continuous-variable quantum key distribution using non-binary low-density parity-check codes, arXiv:1602.09140.
  61. A. G. Mountogiannakis, P. Papanastasiou, B. Braverman, and S. Pirandola, Composably secure data processing for Gaussian-modulated continuous-variable quantum key distribution, Phys. Rev. Res. 4, 013099 (2022).
  62. A. G. Mountogiannakis, P. Papanastasiou, and S. Pirandola, Data postprocessing for the one-way heterodyne protocol under composable finite-size security, Phys. Rev. A 106, 042606 (2022).
  63. W. Weerasinghe, Practical, high-speed, Gaussian modulated coherent state continuous variable quantum key distribution with real-time post processing, Ph.D. thesis, University of Cambridge, 2024.
  64. M. Tomamichel, J. Martinez-Mateo, C. Pacher, and D. Elkouss, Fundamental finite key limits for one-way information reconciliation in quantum key distribution, Quantum Inf. Proc. 16, 280 (2017).
  65. D. J. C. Mackay, Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. Theory 45, 399 (1999).
  66. M. C. Davey and D. MacKay, Low-density parity check codes over GF(q), IEEE Commun. Lett. 2, 165 (1998).
  67. J. Martinez-Mateo and D. Elkouss, Efficient reconciliation of continuous variable quantum key distribution with multiplicatively repeated non-binary LDPC codes, EPJ Quantum Technol. 12, 71 (2025).
  68. F. A. Alaba, M. Othman, I. A. T. Hashem, and F. Alotaibi, Internet of things security: A survey, J. Network Comput. Appl. 88, 10 (2017).
  69. A. Kumar, C. Ottaviani, S. S. Gill, and R. Buyya, Securing the future Internet of things with post-quantum cryptography, Security Privacy 5, e200 (2022).
  70. A. Shafique, S. A. A. Naqvi, A. Raza, M. Ghalaii, P. Papanastasiou, J. McCann, Q. H. Abbasi, and M. A. Imran, A hybrid encryption framework leveraging quantum and classical cryptography for secure transmission of medical images in IoT-based telemedicine networks, Sci. Rep. 14, 31054 (2024).
  71. E. Zadobrischi, The concept regarding vehicular communications based on visible light communication and the IoT, Electronics 12, 1359 (2023).
  72. M. Asif, W. U. Khan, H. M. R. Afzal, J. Nebhen, I. Ullah, A. U. Rehman, and M. K. A. Kabar, Reduced-complexity LDPC decoding for next-generation IoT networks, Wireless Commun. Mobile Comput. 2021, 2029560 (2021).
  73. J. Tang, H. Wen, H.-H. Song, L. Jiao, and K. Zeng, Sharing secrets via wireless broadcasting: A new efficient physical layer group secret key generation for multiple IoT devices, IEEE Internet Things J. 9, 15228 (2022).
  74. X.-H. Tian, R. Yang, H.-Y. Liu, P. Fan, J.-N. Zhang, C. Gu, M. Chen, M. Hu, F.-Y. Lu, C. Zhu, Z.-Q. Yin, Z.-J. Yin, M. Yuan, S. Wang, W. Chen, Y.-X. Gong, S.-N. Zhu, and Z. Xie, Experimental demonstration of drone-based quantum key distribution, Phys. Rev. Lett. 133, 200801 (2024).
  75. O. Elmabrok and M. Razavi, Wireless quantum key distribution in indoor environments, J. Opt. Soc. Am. B 35, 197 (2018).
  76. S. Pirandola, Limits and security of free-space quantum communications, Phys. Rev. Res. 3, 013279 (2021).
  77. In a QKD session, we compute the estimators and their variances directly from the experimental data. These are those effectively used for the worst-case estimators in a practical implementation. For theoretical analysis, one employs asymptotic or large-block approximations. Since in this analysis some of the estimators are a sample mean of independent observations, the central limit theorem ensures approximate Gaussianity, and the Berry-Esseen theorem provides a quantitative bound on the deviation from the Gaussian limit, which for the parameters used in this study (see Table  2) and block sizes 2.5×104–106 is less than 10−2–10−3. Although this bound is conservative, results based on Stein's method and modern Gaussian-approximation theory show that practical convergence is typically faster than the worst-case rate suggests [78, 79, 80]. This supports the Gaussian modeling adopted in our numerical worst-case analysis.
  78. L. H. Y. Chen, L. Goldstein, and Q.-M. Shao, Normal Approximation by Stein's Method (Springer, Berlin, 2011).
  79. M. Tomamichel and V. Y. F. Tan, A tight upper bound for the third-order asymptotics for most discrete memoryless channels, IEEE Trans. Inf. Theory 59, 7041 (2013).
  80. V. Chernozhukov, D. Chetverikov, and K. Kato, Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors, Ann. Stat. 41, 2786 (2013).
  81. D. S. Slepian and J. K. Wolf, Noiseless coding of correlated information sources, IEEE Trans. Inf. Theory 19, 471 (1973).
  82. C. Portmann and R. Renner, Security in quantum cryptography, Rev. Mod. Phys. 94, 025008 (2022).
  83. http://github.com/softquanta/homCVQKD.
  84. http://github.com/eqclabs/tight_bound_leakage.
  85. https://vikingdocs.york.ac.uk/.

Outline

Information

Phys. Rev. A 114, 022611– Published 11 August, 2026
  • Received 8 April 2025
  • Accepted 13 July 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