Imperfect detectors for adversarial tasks with applications to quantum key distribution
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AbstractSecurity analyses in quantum key distribution (QKD) and other adversarial quantum tasks often assume perfect device models. However, real-world implementations often deviate from these models. Thus, it is important to develop security proofs that account for such deviations from ideality. In this work, we extend the idea of squashing maps to develop a general framework for analysing imperfect threshold detectors, treating uncharacterised device parameters such as dark counts and detection efficiencies as adversarially controlled within some ranges. This approach enables a rigorous worst-case analysis with exactly characterised devices, ensuring security proofs remain valid under realistic conditions. Our results strengthen the connection between theoretical security and practical implementations by introducing a flexible framework for integrating detector imperfections into adversarial quantum protocols.Popular summaryReal-world quantum detectors are never perfect—they miss photons and register false clicks. Proving a quantum communication network is secure despite these uncharacterised hardware flaws is notoriously difficult. This paper introduces an elegant theoretical framework that solves this by mathematically packaging these physical glitches into an "untrusted noise channel" given entirely to the eavesdropper. By assuming the adversary controls the hardware's flaws, researchers can drastically simplify their security proofs to act as if the detectors were perfect, bridging the gap between idealised math and practical quantum devices.► BibTeX data@article{Nahar2026imperfectdetectors, doi = {10.22331/q-2026-03-24-2044}, url = {https://doi.org/10.22331/q-2026-03-24-2044}, title = {Imperfect detectors for adversarial tasks with applications to quantum key distribution}, author = {Nahar, Shlok and Tupkary, Devashish and L{\"{u}}tkenhaus, Norbert}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2044}, month = mar, year = {2026} }► References [1] Charles H Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. International Conference on Computers, Systems & Signal Processing 1, pp. 175–179 (1984). url: https://research.ibm.com/publications/quantum-cryptography-public-key-distribution-and-coin-tossing. https://research.ibm.com/publications/quantum-cryptography-public-key-distribution-and-coin-tossing [2] Koenraad MR Audenaert and Martin B Plenio. ``When are correlations quantum?—verification and quantification of entanglement by simple measurements''. New Journal of Physics 8, 266 (2006). https://doi.org/10.1088/1367-2630/8/11/266 [3] Kfir Sulimany, Sri Krishna Vadlamani, Ryan Hamerly, Prahlad Iyengar, and Dirk Englund. ``Quantum-secure multiparty deep learning''. Physical Review X 15, 041056 (2025). https://doi.org/10.1103/k8wg-qmbh [4] Guillermo Currás-Lorenzo, Margarida Pereira, Go Kato, Marcos Curty, and Kiyoshi Tamaki. ``Security framework for quantum key distribution with imperfect sources''. Optica Quantum 3, 525–534 (2025). https://doi.org/10.1364/OPTICAQ.569424 [5] Amir Arqand, Tony Metger, and Ernest Y-Z Tan. ``Mutual information chain rules for security proofs robust against device imperfections'' (2024). url: https://arxiv.org/abs/2407.20396. arXiv:2407.20396 [6] Xoel Sixto, Álvaro Navarrete, Margarida Pereira, Guillermo Currás-Lorenzo, Kiyoshi Tamaki, and Marcos Curty. ``Quantum key distribution with imperfectly isolated devices''. Quantum Science and Technology (2024). https://doi.org/10.1088/2058-9565/addb6e [7] Víctor Zapatero, Álvaro Navarrete, Kiyoshi Tamaki, and Marcos Curty. ``Security of quantum key distribution with intensity correlations''. Quantum 5, 602 (2021). https://doi.org/10.22331/q-2021-12-07-602 [8] Guillermo Currás-Lorenzo, Shlok Nahar, Norbert Lütkenhaus, Kiyoshi Tamaki, and Marcos Curty. ``Security of quantum key distribution with imperfect phase randomisation''. 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Coles, Adam Winick, Jie Lin, and Norbert Lütkenhaus. ``Security proof of practical quantum key distribution with detection-efficiency mismatch''.
Physical Review Research 3, 013076 (2021). https://doi.org/10.1103/PhysRevResearch.3.013076 [18] Toyohiro Tsurumaru and Kiyoshi Tamaki. ``Security proof for quantum-key-distribution systems with threshold detectors''. Phys. Rev. A 78, 032302 (2008). https://doi.org/10.1103/PhysRevA.78.032302 [19] Nicky Kai Hong Li and Norbert Lütkenhaus. ``Improving key rates of the unbalanced phase-encoded BB84 protocol using the flag-state squashing model''.
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Physical Review Applied 20, 064031 (2023). https://doi.org/10.1103/PhysRevApplied.20.064031 [30] Lars Kamin and Norbert Lütkenhaus. ``Improved decoy-state and flag-state squashing methods''. Phys. Rev. Res. 6, 043223 (2024). https://doi.org/10.1103/PhysRevResearch.6.043223 [31] Norbert Lütkenhaus. ``Estimates for practical quantum cryptography''. Phys. Rev. A 59, 3301–3319 (1999). https://doi.org/10.1103/PhysRevA.59.3301 [32] Yanbao Zhang and Norbert Lütkenhaus. ``Entanglement verification with detection-efficiency mismatch''. Physical Review A 95, 042319 (2017). https://doi.org/10.1103/PhysRevA.95.042319 [33] Frédéric Dupuis, Omar Fawzi, and Renato Renner. ``Entropy Accumulation''. Communications in Mathematical Physics 379, 867–913 (2020). https://doi.org/10.1007/s00220-020-03839-5 [34] Tony Metger, Omar Fawzi, David Sutter, and Renato Renner. ``Generalised entropy accumulation''. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS). Pages 844–850. 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In International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings. Page 136. (2004). https://doi.org/10.1109/ISIT.2004.1365172Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-24 15:46:18: Could not fetch cited-by data for 10.22331/q-2026-03-24-2044 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-24 15:46:18: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractSecurity analyses in quantum key distribution (QKD) and other adversarial quantum tasks often assume perfect device models. However, real-world implementations often deviate from these models. Thus, it is important to develop security proofs that account for such deviations from ideality. In this work, we extend the idea of squashing maps to develop a general framework for analysing imperfect threshold detectors, treating uncharacterised device parameters such as dark counts and detection efficiencies as adversarially controlled within some ranges. This approach enables a rigorous worst-case analysis with exactly characterised devices, ensuring security proofs remain valid under realistic conditions. Our results strengthen the connection between theoretical security and practical implementations by introducing a flexible framework for integrating detector imperfections into adversarial quantum protocols.Popular summaryReal-world quantum detectors are never perfect—they miss photons and register false clicks. Proving a quantum communication network is secure despite these uncharacterised hardware flaws is notoriously difficult. This paper introduces an elegant theoretical framework that solves this by mathematically packaging these physical glitches into an "untrusted noise channel" given entirely to the eavesdropper. By assuming the adversary controls the hardware's flaws, researchers can drastically simplify their security proofs to act as if the detectors were perfect, bridging the gap between idealised math and practical quantum devices.► BibTeX data@article{Nahar2026imperfectdetectors, doi = {10.22331/q-2026-03-24-2044}, url = {https://doi.org/10.22331/q-2026-03-24-2044}, title = {Imperfect detectors for adversarial tasks with applications to quantum key distribution}, author = {Nahar, Shlok and Tupkary, Devashish and L{\"{u}}tkenhaus, Norbert}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2044}, month = mar, year = {2026} }► References [1] Charles H Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. International Conference on Computers, Systems & Signal Processing 1, pp. 175–179 (1984). url: https://research.ibm.com/publications/quantum-cryptography-public-key-distribution-and-coin-tossing. https://research.ibm.com/publications/quantum-cryptography-public-key-distribution-and-coin-tossing [2] Koenraad MR Audenaert and Martin B Plenio. ``When are correlations quantum?—verification and quantification of entanglement by simple measurements''. New Journal of Physics 8, 266 (2006). https://doi.org/10.1088/1367-2630/8/11/266 [3] Kfir Sulimany, Sri Krishna Vadlamani, Ryan Hamerly, Prahlad Iyengar, and Dirk Englund. ``Quantum-secure multiparty deep learning''. Physical Review X 15, 041056 (2025). https://doi.org/10.1103/k8wg-qmbh [4] Guillermo Currás-Lorenzo, Margarida Pereira, Go Kato, Marcos Curty, and Kiyoshi Tamaki. ``Security framework for quantum key distribution with imperfect sources''. Optica Quantum 3, 525–534 (2025). https://doi.org/10.1364/OPTICAQ.569424 [5] Amir Arqand, Tony Metger, and Ernest Y-Z Tan. ``Mutual information chain rules for security proofs robust against device imperfections'' (2024). url: https://arxiv.org/abs/2407.20396. arXiv:2407.20396 [6] Xoel Sixto, Álvaro Navarrete, Margarida Pereira, Guillermo Currás-Lorenzo, Kiyoshi Tamaki, and Marcos Curty. ``Quantum key distribution with imperfectly isolated devices''. Quantum Science and Technology (2024). https://doi.org/10.1088/2058-9565/addb6e [7] Víctor Zapatero, Álvaro Navarrete, Kiyoshi Tamaki, and Marcos Curty. ``Security of quantum key distribution with intensity correlations''. Quantum 5, 602 (2021). https://doi.org/10.22331/q-2021-12-07-602 [8] Guillermo Currás-Lorenzo, Shlok Nahar, Norbert Lütkenhaus, Kiyoshi Tamaki, and Marcos Curty. ``Security of quantum key distribution with imperfect phase randomisation''. Quantum Science and Technology 9, 015025 (2023). https://doi.org/10.1088/2058-9565/ad141c [9] Chi-hang Fred Fung, Kiyoshi Tamaki, Bing Qi, Hoi-Kwong Lo, and Xiongfeng Ma. ``Security proof of quantum key distribution with detection efficiency mismatch'' (2008). url: https://arxiv.org/abs/0802.3788. arXiv:0802.3788 [10] Lars Lydersen and Johannes Skaar. ``Security of quantum key distribution with bit and basis dependent detector flaws'' (2010). arxiv:0807.0767. arXiv:0807.0767 [11] Øystein Marøy, Lars Lydersen, and Johannes Skaar. ``Security of quantum key distribution with arbitrary individual imperfections''. Phys. Rev. A 82, 032337 (2010). https://doi.org/10.1103/PhysRevA.82.032337 [12] Devashish Tupkary, Shlok Nahar, Pulkit Sinha, and Norbert Lütkenhaus. ``Phase error rate estimation in QKD with imperfect detectors''. Quantum 9, 1937 (2025). https://doi.org/10.22331/q-2025-12-11-1937 [13] Ashutosh Marwah and Frédéric Dupuis. ``Proving security of BB84 under source correlations'' (2024). url: https://arxiv.org/abs/2402.12346. arXiv:2402.12346 [14] Toyohiro Tsurumaru. ``Squash operator and symmetry''. Phys. Rev. A 81, 012328 (2010). https://doi.org/10.1103/PhysRevA.81.012328 [15] 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 [16] O. Gittsovich, N. J. Beaudry, V. Narasimhachar, R. Romero Alvarez, T. Moroder, and N. Lütkenhaus. ``Squashing model for detectors and applications to quantum-key-distribution protocols''. Phys. Rev. A 89, 012325 (2014). https://doi.org/10.1103/PhysRevA.89.012325 [17] Yanbao Zhang, Patrick J. Coles, Adam Winick, Jie Lin, and Norbert Lütkenhaus. ``Security proof of practical quantum key distribution with detection-efficiency mismatch''.
Physical Review Research 3, 013076 (2021). https://doi.org/10.1103/PhysRevResearch.3.013076 [18] Toyohiro Tsurumaru and Kiyoshi Tamaki. ``Security proof for quantum-key-distribution systems with threshold detectors''. Phys. Rev. A 78, 032302 (2008). https://doi.org/10.1103/PhysRevA.78.032302 [19] Nicky Kai Hong Li and Norbert Lütkenhaus. ``Improving key rates of the unbalanced phase-encoded BB84 protocol using the flag-state squashing model''.
Physical Review Research 2, 043172 (2020). https://doi.org/10.1103/PhysRevResearch.2.043172 [20] Amir Arqand and Ernest Y-Z Tan. ``Marginal-constrained entropy accumulation theorem'' (2025). url: https://arxiv.org/abs/2502.02563. arXiv:2502.02563 [21] Zhiyao Wang, Devashish Tupkary, and Shlok Nahar. ``Phase error estimation for passive detection setups with imperfections and memory effects'' (2025). url: https://arxiv.org/abs/2508.21486. arXiv:2508.21486 [22] Thu Ha Dao, Francesco Amanti, Greta Andrini, Fabrizio Armani, Fabrizio Barbato, Vittorio Bellani, Vincenzo Bonaiuto, Simone Cammarata, Matteo Campostrini, Samuele Cornia, et al. ``Single-photon detectors for quantum integrated photonics''. Photonics 12, 8 (2024). https://doi.org/10.3390/photonics12010008 [23] Alan Migdall, Sergey V Polyakov, Jingyun Fan, and Joshua C Bienfang. ``Single-photon generation and detection: physics and applications''. Academic Press. (2013). url: https://www.sciencedirect.com/bookseries/experimental-methods-in-the-physical-sciences/vol/45. https://www.sciencedirect.com/bookseries/experimental-methods-in-the-physical-sciences/vol/45 [24] Tobias Moroder, Otfried Gühne, Normand Beaudry, Marco Piani, and Norbert Lütkenhaus. ``Entanglement verification with realistic measurement devices via squashing operations''. Physical Review A 81, 052342 (2010). https://doi.org/10.1103/PhysRevA.81.052342 [25] Shlok Nahar, Devashish Tupkary, Yuming Zhao, Norbert Lütkenhaus, and Ernest Y.-Z. Tan. ``Postselection technique for optical quantum key distribution with improved de finetti reductions''. PRX Quantum 5, 040315 (2024). https://doi.org/10.1103/PRXQuantum.5.040315 [26] Masato Koashi. ``Efficient quantum key distribution with practical sources and detectors'' (2006). url: https://arxiv.org/abs/quant-ph/0609180. arXiv:quant-ph/0609180 [27] Varun Narasimhachar. ``Study of realistic devices for quantum key-distribution''. Master's thesis. University of Waterloo. (2011). url: https://uwspace.uwaterloo.ca/items/ad80136b-910d-4211-b5fe-dfcf49b03cd4. https://uwspace.uwaterloo.ca/items/ad80136b-910d-4211-b5fe-dfcf49b03cd4 [28] Nicky Kai Hong Li. ``Application of the Flag-State Squashing Model to Numerical Quantum Key Distribution Security Analysis''. Master's thesis. University of Waterloo. (2020). url: https://uwspace.uwaterloo.ca/handle/10012/16320. https://uwspace.uwaterloo.ca/handle/10012/16320 [29] Shlok Nahar, Twesh Upadhyaya, and Norbert Lütkenhaus. ``Imperfect phase randomization and generalized decoy-state quantum key distribution''.
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In International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings. Page 136. (2004). https://doi.org/10.1109/ISIT.2004.1365172Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-24 15:46:18: Could not fetch cited-by data for 10.22331/q-2026-03-24-2044 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-24 15:46:18: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.