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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.<\/jats:p>","DOI":"10.22331\/q-2025-12-11-1937","type":"journal-article","created":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T12:37:56Z","timestamp":1765456676000},"page":"1937","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":12,"title":["Phase error rate estimation in QKD with imperfect detectors"],"prefix":"10.22331","volume":"9","author":[{"given":"Devashish","family":"Tupkary","sequence":"first","affiliation":[{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shlok","family":"Nahar","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pulkit","family":"Sinha","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing and School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Norbert","family":"L\u00fctkenhaus","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2025,12,11]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Marco Tomamichel and Renato Renner. ``Uncertainty Relation for Smooth Entropies''. 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