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Phase error rate estimation in QKD with imperfect detectors

Devashish Tupkary, Shlok Nahar, Pulkit Sinha, and Norbert Lütkenhaus

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Researchers developed a finite-size security proof for decoy-state BB84 quantum key distribution (QKD) that accommodates imperfect detectors, using entropic uncertainty relations to bound phase error rates without modifying existing protocols or hardware. The proof addresses basis-efficiency mismatch in detectors by requiring only the maximum relative difference in detection efficiencies and dark count rates, eliminating the need for full characterization of each detector’s imperfections. It accounts for detector side-channel vulnerabilities by allowing an eavesdropper to dynamically adjust detector parameters within known ranges during each round, strengthening security against adaptive attacks. The framework supports variable-length key generation, enabling users to adaptively determine key length and error-correction bits based on real-time protocol observations, enhancing practical flexibility. Quantitative analysis demonstrates the impact of basis-efficiency mismatch on decoy-state BB84, providing concrete bounds for secure implementation in real-world QKD systems.

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AbstractWe present a finite-size security proof of the decoy-state BB84 QKD protocol against coherent attacks, using entropic uncertainty relations, for imperfect detectors. We apply this result to the case of detectors with imperfectly characterized basis-efficiency mismatch. Our proof works by obtaining a suitable bound on the phase error rate, without requiring any new modifications to the protocol steps or hardware. It is applicable to imperfectly characterized detectors, and only requires the maximum relative difference in detection efficiencies and dark count rates of the detectors to be characterized. Moreover, our proof allows Eve to choose detector efficiencies and dark count rates in their allowed ranges in each round, thereby addressing an important problem of detector side channels. We prove security in the variable-length framework, where users are allowed to adaptively determine the length of key to be produced, and number of bits to be used for error-correction, based on observations made during the protocol. We quantitatively demonstrate the effect of basis-efficiency mismatch by applying our results to the decoy-state BB84 protocol. ► BibTeX data@article{Tupkary2025phaseerrorrate, doi = {10.22331/q-2025-12-11-1937}, url = {https://doi.org/10.22331/q-2025-12-11-1937}, title = {Phase error rate estimation in {QKD} with imperfect detectors}, author = {Tupkary, Devashish and Nahar, Shlok and Sinha, Pulkit 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 = {1937}, month = dec, year = {2025} }► References [1] Marco Tomamichel and Renato Renner. ``Uncertainty Relation for Smooth Entropies''.

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Springer International Publishing. Cham (2016). https://doi.org/10.1007/978-3-319-21891-5 [63] Marco Tomamichel. ``A Framework for Non-Asymptotic Quantum Information Theory'' (2013). arXiv:1203.2142. arXiv:1203.2142 [64] Wassily Hoeffding. ``Probability Inequalities for Sums of Bounded Random Variables''. Journal of the American Statistical Association 58, 13–30 (1963). https://doi.org/10.2307/2282952Cited byOn Crossref's cited-by service no data on citing works was found (last attempt 2025-12-28 13:25:42). Could not fetch ADS cited-by data during last attempt 2025-12-28 13:25:43: 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. AbstractWe present a finite-size security proof of the decoy-state BB84 QKD protocol against coherent attacks, using entropic uncertainty relations, for imperfect detectors. We apply this result to the case of detectors with imperfectly characterized basis-efficiency mismatch. Our proof works by obtaining a suitable bound on the phase error rate, without requiring any new modifications to the protocol steps or hardware. It is applicable to imperfectly characterized detectors, and only requires the maximum relative difference in detection efficiencies and dark count rates of the detectors to be characterized. Moreover, our proof allows Eve to choose detector efficiencies and dark count rates in their allowed ranges in each round, thereby addressing an important problem of detector side channels. We prove security in the variable-length framework, where users are allowed to adaptively determine the length of key to be produced, and number of bits to be used for error-correction, based on observations made during the protocol. We quantitatively demonstrate the effect of basis-efficiency mismatch by applying our results to the decoy-state BB84 protocol. ► BibTeX data@article{Tupkary2025phaseerrorrate, doi = {10.22331/q-2025-12-11-1937}, url = {https://doi.org/10.22331/q-2025-12-11-1937}, title = {Phase error rate estimation in {QKD} with imperfect detectors}, author = {Tupkary, Devashish and Nahar, Shlok and Sinha, Pulkit 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 = {1937}, month = dec, year = {2025} }► References [1] Marco Tomamichel and Renato Renner. ``Uncertainty Relation for Smooth Entropies''.

Physical Review Letters 91, 057901 (2003). https://doi.org/10.1103/PhysRevLett.91.057901 [43] Hoi-Kwong Lo, Xiongfeng Ma, and Kai Chen. ``Decoy State Quantum Key Distribution''.

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Phase error rate estimation in QKD with imperfect detectors

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