{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:56:30Z","timestamp":1760234190884,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,23]],"date-time":"2021-04-23T00:00:00Z","timestamp":1619136000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2020YFA0309702"],"award-info":[{"award-number":["2020YFA0309702"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61605248","61675235","61505261"],"award-info":[{"award-number":["61605248","61675235","61505261"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Henan","award":["202300410534","202300410532"],"award-info":[{"award-number":["202300410534","202300410532"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The twin-field quantum key distribution (TF-QKD) protocol and its variations have been proposed to overcome the linear Pirandola\u2013Laurenza\u2013Ottaviani\u2013Banchi (PLOB) bound. One variation called phase-matching QKD (PM-QKD) protocol employs discrete phase randomization and the phase post-compensation technique to improve the key rate quadratically. However, the discrete phase randomization opens a loophole to threaten the actual security. In this paper, we first introduce the unambiguous state discrimination (USD) measurement and the photon-number-splitting (PNS) attack against PM-QKD with imperfect phase randomization. Then, we prove the rigorous security of decoy state PM-QKD with discrete phase randomization. Simulation results show that, considering the intrinsic bit error rate and sifting factor, there is an optimal discrete phase randomization value to guarantee security and performance. Furthermore, as the number of discrete phase randomization increases, the key rate of adopting vacuum and one decoy state approaches infinite decoy states, the key rate between discrete phase randomization and continuous phase randomization is almost the same.<\/jats:p>","DOI":"10.3390\/e23050508","type":"journal-article","created":{"date-parts":[[2021,4,23]],"date-time":"2021-04-23T12:08:30Z","timestamp":1619179710000},"page":"508","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Phase-Matching Quantum Key Distribution with Discrete Phase Randomization"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2789-5938","authenticated-orcid":false,"given":"Xiaoxu","family":"Zhang","sequence":"first","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Basic Department, SSF IEU, Zhengzhou 450001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9774-1064","authenticated-orcid":false,"given":"Yang","family":"Wang","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Musheng","family":"Jiang","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8212-0556","authenticated-orcid":false,"given":"Yifei","family":"Lu","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongwei","family":"Li","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chun","family":"Zhou","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wansu","family":"Bao","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,23]]},"reference":[{"key":"ref_1","unstructured":"Bennett, C.H., and Brassard, G. 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