Finite-Key Analysis of Quantum Key Distribution with Characterized Devices Using Entropy Accumulation

Ian George; Jie Lin; Thomas van Himbeeck; Kun Fang; Norbert Lütkenhaus

doi:10.22331/q-2025-12-12-1941

Finite-Key Analysis of Quantum Key Distribution with Characterized Devices Using Entropy Accumulation

Ian George^1,2, Jie Lin¹, Thomas van Himbeeck^1,3, Kun Fang^1,4, and Norbert Lütkenhaus¹

¹Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
²Department of Electrical & Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA
³Department of Electrical & Computer Engineering, University of Toronto, Toronto, Ontario M5S 3G4, Canada
⁴School of Data Science, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, China

Published:	2025-12-12, volume 9, page 1941
Editor:	Anna Pappa
Eprint:	arXiv:2203.06554v2
Doi:	https://doi.org/10.22331/q-2025-12-12-1941
Citation:	Quantum 9, 1941 (2025).

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Abstract

The Entropy Accumulation Theorem (EAT) was introduced to significantly improve the finite-size rates for device-independent quantum information processing tasks such as device-independent quantum key distribution (QKD). A natural question would be whether it also improves the rates for device-dependent QKD. In this work, we provide an affirmative answer to this question. We present new tools for applying the EAT in the device-dependent setting. We present sufficient conditions for the Markov chain conditions to hold as well as general algorithms for constructing the needed min-tradeoff function. Utilizing Dupuis' recent privacy amplification without smoothing result, we improve the key rate by optimizing the sandwiched Rényi entropy directly rather than considering the traditional smooth min-entropy. We exemplify these new tools by considering several examples including the BB84 protocol with the qubit-based version and with a realistic parametric down-conversion source, the six-state four-state protocol and a high-dimensional analog of the BB84 protocol.

► BibTeX data

@article{George2025finitekeyanalysisof,
  doi = {10.22331/q-2025-12-12-1941},
  url = {https://doi.org/10.22331/q-2025-12-12-1941},
  title = {Finite-{K}ey {A}nalysis of {Q}uantum {K}ey {D}istribution with {C}haracterized {D}evices {U}sing {E}ntropy {A}ccumulation},
  author = {George, Ian and Lin, Jie and van Himbeeck, Thomas and Fang, Kun and L{\"{u}}tkenhaus, Norbert},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1941},
  month = dec,
  year = {2025}
}

► References

[1] Charles H. Bennett and Gilles Brassard. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, pages 175–179, New York, 1984. IEEE. https://doi.org/10.1016/j.tcs.2014.05.025.
https://doi.org/10.1016/j.tcs.2014.05.025

[2] Artur K. Ekert. Quantum cryptography based on Bell's theorem. Phys. Rev. Lett., 67 (6): 661, 1991. https://doi.org/10.1103/PhysRevLett.67.661.
https://doi.org/10.1103/PhysRevLett.67.661

[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. https://doi.org/10.1103/RevModPhys.81.1301.
https://doi.org/10.1103/RevModPhys.81.1301

[4] Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan. Secure quantum key distribution with realistic devices. Rev. Mod. Phys., 92 (2): 025002, 2020. https://doi.org/10.1103/RevModPhys.92.025002.
https://doi.org/10.1103/RevModPhys.92.025002

[5] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. 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: 1012–1236, 2020. https://doi.org/10.1364/AOP.361502.
https://doi.org/10.1364/AOP.361502

[6] Alberto Boaron, Gianluca Boso, Davide Rusca, Cédric Vulliez, Claire Autebert, Misael Caloz, Matthieu Perrenoud, Gaëtan Gras, Félix Bussières, Ming-Jun Li, Daniel Nolan, Anthony Martin, and Hugo Zbinden. Secure quantum key distribution over 421 km of optical fiber. Phys. Rev. Lett., 121 (19): 190502, 2018. https://doi.org/10.1103/PhysRevLett.121.190502.
https://doi.org/10.1103/PhysRevLett.121.190502

[7] Xiao-Tian Fang, Pei Zeng, Hui Liu, Mi Zou, Weijie Wu, Yan-Lin Tang, Ying-Jie Sheng, Yao Xiang, Weijun Zhang, Hao Li, Zhen Wang, Lixing You, Hao Chen Ming-Jun Li, Yu-Ao Chen, Qiang Zhang, Cheng-Zhi Peng, Xiongfeng Ma, Teng-Yun Chen, and Jian-Wei Pan. Implementation of quantum key distribution surpassing the linear rate-transmittance bound. Nat. Photonics, 14 (7): 422–425, 2020. https://doi.org/10.1038/s41566-020-0599-8.
https://doi.org/10.1038/s41566-020-0599-8

[8] Sheng-Kai Liao, Wen-Qi Cai, Wei-Yue Liu, Liang Zhang, Yang Li, Ji-Gang Ren, Juan Yin, Qi Shen, Yuan Cao, Zheng-Ping Li, Feng-Zhi Li, Xia-Wei Chen, Li-Hua Sun, Jian-Jun Jia, Jin-Cai Wu, Xiao-Jun Jiang, Jian-Feng Wang, Yong-Mei Huang, Qiang Wang, Yi-Lin Zhou, Lei Deng, Tao Xi, Lu Ma, Tai Hu, Qiang Zhang, Yu-Ao Chen, Nai-Le Liu, Xiang-Bin Wang, Zhen-Cai Zhu, Chao-Yang Lu, Rong Shu, Cheng-Zhi Peng, Jian-Yu Wang, and Jian-Wei Pan. Satellite-to-ground quantum key distribution. Nature, 549 (7670): 43–47, 2017. https://doi.org/10.1038/nature23655.
https://doi.org/10.1038/nature23655

[9] Robert Bedington, Juan Miguel Arrazola, and Alexander Ling. Progress in satellite quantum key distribution. npj Quantum Inf., 3 (1): 1–13, 2017. https://doi.org/10.1038/s41534-017-0031-5.
https://doi.org/10.1038/s41534-017-0031-5

[10] P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. O’Brien, and M. G. Thompson. Chip-based quantum key distribution. Nat. Commun., 8 (1): 13984, 2017. https://doi.org/10.1038/ncomms13984.
https://doi.org/10.1038/ncomms13984

[11] G. Zhang, J. Y. Haw, H. Cai, F. Xu, S. M. Assad, J. F. Fitzsimons, X. Zhou, Y. Zhang, S Yu, J Wu, W. Ser, L. C. Kwek, and A. Q. Liu. An integrated silicon photonic chip platform for continuous-variable quantum key distribution. Nat. Photonics, 13 (12): 839–842, 2019. https://doi.org/10.1038/s41566-019-0504-5.
https://doi.org/10.1038/s41566-019-0504-5

[12] Kejin Wei, Wei Li, Hao Tan, Yang Li, Hao Min, Wei-Jun Zhang, Hao Li, Lixing You, Zhen Wang, Xiao Jiang, Teng-Yun Chen, Sheng-Kai Liao, Cheng-Zhi Peng, Feihu Xu, and Jian-Wei Pan. High-speed measurement-device-independent quantum key distribution with integrated silicon photonics. Phys. Rev. X, 10 (3): 031030, 2020. https://doi.org/10.1103/PhysRevX.10.031030.
https://doi.org/10.1103/PhysRevX.10.031030

[13] Renato Renner. Security of quantum key distribution. International Journal of Quantum Inforemation, 6: 1–127, 2005. https://doi.org/10.1142/S0219749908003256.
https://doi.org/10.1142/S0219749908003256

[14] Valerio Scarani and Renato Renner. Security bounds for quantum cryptography with finite resources. In Y. Kawano and M. Mosca, editors, Theory of Quantum Computation, Communication, and Cryptography, pages 85–95, Berlin, Heidelberg, 2008. Springer. https://doi.org/10.1007/978-3-540-89304-2_8.
https://doi.org/10.1007/978-3-540-89304-2_8

[15] Matthias Christandl, Robert König, and Renato Renner. Postselection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett., 102: 020504, 2009. https://doi.org/10.1103/PhysRevLett.102.020504.
https://doi.org/10.1103/PhysRevLett.102.020504

[16] Frederic Dupuis, Omar Fawzi, and Renato Renner. Entropy accumulation. Commun. Math. Phys., 379: 867–913, 2020. https://doi.org/10.1007/s00220-020-03839-5.
https://doi.org/10.1007/s00220-020-03839-5

[17] Frédéric Dupuis and Omar Fawzi. Entropy accumulation with improved second-order term. IEEE Trans. Inf. Theory, 65: 7596–7612, 2019. https://doi.org/10.1109/TIT.2019.2929564.
https://doi.org/10.1109/TIT.2019.2929564

[18] Rotem Arnon-Friedman, Frédéric Dupuis, Omar Fawzi, Renato Renner, and Thomas Vidick. Practical device-independent quantum cryptography via entropy accumulation. Nature communications, 9 (1): 459, 2018. https://doi.org/10.1038/s41467-017-02307-4.
https://doi.org/10.1038/s41467-017-02307-4

[19] Rotem Arnon-Friedman, Renato Renner, and Thomas Vidick. Simple and tight device-independent security proofs. SIAM J. Comput., 48 (1): 181–225, 2019. https://doi.org/10.1137/18M1174726.
https://doi.org/10.1137/18M1174726

[20] D. P. Nadlinger, P. Drmota, B. C. Nichol, G. Araneda, D. Main, R. Srinivas, D. M. Lucas, C. J. Ballance, K. Ivanov, E. Y-Z. Tan, P. Sekatski, R. L. Urbanke, R. Renner, N. Sangouard, and J-D. Bancal. Experimental quantum key distribution certified by bell's theorem. Nature, 607: 682–686, 2022. https://doi.org/10.1038/s41586-022-04941-5.
https://doi.org/10.1038/s41586-022-04941-5

[21] Wei Zhang, Tim van Leent, Kai Redeker, Robert Garthoff, Rene Schwonnek, Florian Fertig, Sebastian Eppelt, Valerio Scarani, Charles C. W. Lim, and Harald Weinfurter. A device-independent quantum key distribution system for distant users. Nature, 607: 687–691, 2022. https://doi.org/10.1038/s41586-022-04891-y.
https://doi.org/10.1038/s41586-022-04891-y

[22] Wen-Zhao Liu, Yu-Zhe Zhang, Yi-Zheng Zhen, Ming-Han Li, Yang Liu, Jingyun Fan, Feihu Xu, Qiang Zhang, and Jian-Wei Pan. Toward a photonic demonstration of device-independent quantum key distribution. Phys. Rev. Lett., 129: 050502, Jul 2022. https://doi.org/10.1103/PhysRevLett.129.050502.
https://doi.org/10.1103/PhysRevLett.129.050502

[23] Jonathan Barrett, Roger Colbeck, and Adrian Kent. Memory attacks on device-independent quantum cryptography. Phys. Rev. Lett., 110: 010503, Jan 2013. https://doi.org/10.1103/PhysRevLett.110.010503.
https://doi.org/10.1103/PhysRevLett.110.010503

[24] Tony Metger, Omar Fawzi, David Sutter, and Renato Renner. Generalised entropy accumulation. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), page 844–850. IEEE, oct 2022. https://doi.org/10.1109/focs54457.2022.00085.
https://doi.org/10.1109/focs54457.2022.00085

[25] Tony Metger and Renato Renner. Security of quantum key distribution from generalised entropy accumulation. Nature Communications, 14 (1), August 2023. ISSN 2041-1723. https://doi.org/10.1038/s41467-023-40920-8.
https://doi.org/10.1038/s41467-023-40920-8

[26] Peter Brown, Hamza Fawzi, and Omar Fawzi. Computing conditional entropies for quantum correlations. Nat. Commun., 12: 575, 2021. https://doi.org/10.1038/s41467-020-20018-1.
https://doi.org/10.1038/s41467-020-20018-1

[27] Patrick J. Coles, Eric M. Metodiev, and Norbert Lütkenhaus. Numerical approach for unstructured quantum key distribution. Nat. Commun., 7: 11712, 2016. https://doi.org/10.1038/ncomms11712.
https://doi.org/10.1038/ncomms11712

[28] Adam Winick, Norbert Lütkenhaus, and Patrick J. Coles. Reliable numerical key rates for quantum key distribution. Quantum, 2: 77, 2018. https://doi.org/10.22331/q-2018-07-26-77.
https://doi.org/10.22331/q-2018-07-26-77

[29] Frédéric Dupuis. Privacy amplification and decoupling without smoothing. IEEE Transactions on Information Theory, 69 (12): 7784–7792, 2023. https://doi.org/10.1109/TIT.2023.3301812.
https://doi.org/10.1109/TIT.2023.3301812

[30] Silvestre Abruzzo, Hermann Kampermann, Markus Mertz, and Dagmar Bruß. Quantum key distribution with finite resources: Secret key rates via Rényi entropies. Physical Review A, 84 (3), September 2011. ISSN 1094-1622. https://doi.org/10.1103/physreva.84.032321.
https://doi.org/10.1103/physreva.84.032321

[31] Masahito Hayashi. Large deviation analysis for quantum security via smoothing of Rényi entropy of order 2. IEEE Transactions on Information Theory, 60 (10): 6702–6732, October 2014. ISSN 1557-9654. https://doi.org/10.1109/TIT.2014.2337884.
https://doi.org/10.1109/TIT.2014.2337884

[32] Ramy Tannous, Zhangdong Ye, Jeongwan Jin, Katanya B. Kuntz, Norbert Lütkenhaus, and Thomas Jennewein. Demonstration of a 6 state-4 state reference frame independent channel for quantum key distribution. Appl. Phys. Lett., 115: 211103, 2019. https://doi.org/10.1063/1.5125700.
https://doi.org/10.1063/1.5125700

[33] Agnes Ferenczi and Norbert Lütkenhaus. Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning. Phys. Rev. A, 85: 052310, 2012. https://doi.org/10.1103/PhysRevA.85.052310.
https://doi.org/10.1103/PhysRevA.85.052310

[34] Marco Tomamichel. Quantum Information Processing with Finite Resources. Springer International Publishing, 2016. https://doi.org/10.1007/978-3-319-21891-5. All equation numbers and theorem numbers cited in this work refer to the fourth arXiv version of this cited work: https://arxiv.org/abs/1504.00233v4.
https://doi.org/10.1007/978-3-319-21891-5
arXiv:1504.00233v4

[35] Marco Tomamichel. A Framework for Non-Asymptotic Quantum Infomration Theory. PhD thesis, ETH Zürich, Zürich, Switzerland, 2012. URL https://arxiv.org/abs/1203.2142.
arXiv:1203.2142

[36] Christopher Portmann and Renato Renner. Cryptographic security of quantum key distribution. arXiv preprint arXiv:1409.3525, 2014. https://doi.org/10.48550/arXiv.1409.3525.
https://doi.org/10.48550/arXiv.1409.3525
arXiv:1409.3525

[37] David Sutter. Approximate quantum Markov chains. In Approximate Quantum Markov Chains, pages 75–100. Springer, 2018. https://doi.org/10.1007/978-3-319-78732-9_5.
https://doi.org/10.1007/978-3-319-78732-9_5

[38] Igor Devetak and Andreas Winter. Distillation of secret key entanglement from quantum states. Proc. R. Soc. A, 461: 207–235, 2005. https://doi.org/10.1098/rspa.2004.1372.
https://doi.org/10.1098/rspa.2004.1372

[39] Rotem Arnon-Friedman. Device-Independent Quantum Information Processing: A Simplified Analysis. Springer Nature, 2020. https://doi.org/10.1007/978-3-030-60231-4.
https://doi.org/10.1007/978-3-030-60231-4

[40] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory, Second Edition. John Wiley & Sons, 2005. ISBN 0471241954. https://doi.org/10.1002/047174882X.
https://doi.org/10.1002/047174882X

[41] Ian George. Numerical finite key analysis. Master's thesis, University of Waterloo, 2020.

[42] Ian George, Jie Lin, and Norbert Lütkenhaus. Numerical calculations of the finite key rate for general quantum key distribution protocols. Phys. Rev. Research, 3: 013274, 2021. https://doi.org/10.1103/PhysRevResearch.3.013274.
https://doi.org/10.1103/PhysRevResearch.3.013274

[43] John Watrous. The Theory of Quantum Information. Cambridge University Press, Cambridge, UK, 2018. ISBN 1107180562. https://doi.org/10.1017/9781316848142.
https://doi.org/10.1017/9781316848142

[44] Hoi-Kwong Lo, H. F. Chau, and M. Ardehali. Efficient quantum key distribution scheme and a proof of its unconditional security. J. Cryptol., 18: 133–165, 2005. https://doi.org/10.1007/s00145-004-0142-y.
https://doi.org/10.1007/s00145-004-0142-y

[45] Anthony Laing, Valerio Scarani, John G. Rarity, and Jeremy L. O’Brien. Reference-frame-independent quantum key distribution. Phys. Rev. A, 82 (3): 012304, 2010. https://doi.org/10.1103/PhysRevA.82.012304.
https://doi.org/10.1103/PhysRevA.82.012304

[46] Barbara Kraus, Nicolas Gisin, and Renato Renner. Lower and upper bounds on the secret-key rate for quantum key distribution protocols using one-way classical communication. Phys. Rev. Lett., 95 (8): 080501, 2005. https://doi.org/10.1103/PhysRevLett.95.080501.
https://doi.org/10.1103/PhysRevLett.95.080501

[47] Renato Renner, Nicolas Gisin, and Barbara Kraus. Information-theoretic security proof for quantum-key-distribution protocols. Phys. Rev. A, 72 (1): 012332, 2005. https://doi.org/10.1103/PhysRevA.72.012332.
https://doi.org/10.1103/PhysRevA.72.012332

[48] Lana Sheridan and Valerio Scarani. Security proof for quantum key distribution using qudit systems. Phys. Rev. A, 82 (3): 030301(R), 2010. https://doi.org/10.1103/PhysRevA.82.030301.
https://doi.org/10.1103/PhysRevA.82.030301

[49] Pieter Kok and Samuel L. Braunstein. Postselected versus nonpostselected quantum teleportation using parametric down-conversion. Phys. Rev. A, 61 (4): 042304, 2000. https://doi.org/10.1103/PhysRevA.61.042304.
https://doi.org/10.1103/PhysRevA.61.042304

[50] Xiongfeng Ma, Chi-Hang Fred Fung, and Hoi-Kwong Lo. Quantum key distribution with entangled photon sources. Phys. Rev. A, 76: 012307, 2007. https://doi.org/10.1103/PhysRevA.76.012307.
https://doi.org/10.1103/PhysRevA.76.012307

[51] Normand J. Beaudry, Tobias Moroder, and Norbert Lütkenhaus. Squashing models for optical measurements in quantum communication. Phys. Rev. Lett., 101: 093601, 2008. https://doi.org/10.1103/PhysRevLett.101.093601.
https://doi.org/10.1103/PhysRevLett.101.093601

[52] O. Gittsovich, N. J. Beaudry, V. Narasimhachar, R. R. Alvarez, T. Moroder, and N. Lütkenhaus. Squashing models for detectors and applications to quantum key distribution protocols. Phys. Rev. A, 89: 012325, 2014. https://doi.org/10.1103/PhysRevA.89.012325.
https://doi.org/10.1103/PhysRevA.89.012325

[53] Marcos Curty, Maciej Lewenstein, and Norbert Lütkenhaus. Entanglement as precondition for secure quantum key distribution. Phys. Rev. Lett., 92: 217903, 2004. https://doi.org/10.1103/PhysRevLett.92.217903.
https://doi.org/10.1103/PhysRevLett.92.217903

[54] Lars Kamin, Amir Arqand, Ian George, Norbert Lütkenhaus, and Ernest Y.-Z. Tan. Finite-size analysis of prepare-and-measure and decoy-state quantum key distribution via entropy accumulation. PRX Quantum, 6: 020342, Jun 2025. https://doi.org/10.1103/PRXQuantum.6.020342.
https://doi.org/10.1103/PRXQuantum.6.020342

[55] Frédéric Dupuis. Chain rules for quantum Rényi entropies. Journal of Mathematical Physics, 56 (2): 022203, 2015. https://doi.org/10.1063/1.4907981.
https://doi.org/10.1063/1.4907981

[56] Marco Tomamichel and Anthony Leverrier. A largely self-contained and complete security proof for quantum key distribution. Quantum, 1: 14, 2017. https://doi.org/10.22331/q-2017-07-14-14.
https://doi.org/10.22331/q-2017-07-14-14

[57] Dan Romik. Stirling's approximation for n!: the ultimate short proof? The American Mathematical Monthly, 107 (6): 556–557, 2000. https://doi.org/10.1080/00029890.2000.12005235.
https://doi.org/10.1080/00029890.2000.12005235

[58] Jonathan Borwein and Adrian S. Lewis. Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer-Verlag, New York, USA, 2006. https://doi.org/10.1007/978-0-387-31256-9.
https://doi.org/10.1007/978-0-387-31256-9

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