{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T15:32:39Z","timestamp":1772724759162,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T00:00:00Z","timestamp":1675123200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Key Research and Development Program of China","award":["2020YFA0309802"],"award-info":[{"award-number":["2020YFA0309802"]}]},{"name":"the National Key Research and Development Program of China","award":["62171424"],"award-info":[{"award-number":["62171424"]}]},{"name":"the National Key Research and Development Program of China","award":["61961136004"],"award-info":[{"award-number":["61961136004"]}]},{"name":"the National Natural Science Foundation of China","award":["2020YFA0309802"],"award-info":[{"award-number":["2020YFA0309802"]}]},{"name":"the National Natural Science Foundation of China","award":["62171424"],"award-info":[{"award-number":["62171424"]}]},{"name":"the National Natural Science Foundation of China","award":["61961136004"],"award-info":[{"award-number":["61961136004"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Quantum key distribution (QKD) allows two remote parties to share information-theoretic secret keys. Many QKD protocols assume the phase of encoding state can be continuous randomized from 0 to 2\u03c0, which, however, may be questionable in the experiment. This is particularly the case in the recently proposed twin-field (TF) QKD, which has received a lot of attention since it can increase the key rate significantly and even beat some theoretical rate-loss limits. As an intuitive solution, one may introduce discrete-phase randomization instead of continuous randomization. However, a security proof for a QKD protocol with discrete-phase randomization in the finite-key region is still missing. Here, we develop a technique based on conjugate measurement and quantum state distinguishment to analyze the security in this case. Our results show that TF-QKD with a reasonable number of discrete random phases, e.g., 8 phases from {0,\u03c0\/4,\u03c0\/2,\u2026,7\u03c0\/4}, can achieve satisfactory performance. On the other hand, we find the finite-size effects become more notable than before, which implies that more pulses should be emit in this case. More importantly, as a the first proof for TF-QKD with discrete-phase randomization in the finite-key region, our method is also applicable in other QKD protocols.<\/jats:p>","DOI":"10.3390\/e25020258","type":"journal-article","created":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T02:29:03Z","timestamp":1675218543000},"page":"258","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Finite-Key Analysis for Quantum Key Distribution with Discrete-Phase Randomization"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5296-6507","authenticated-orcid":false,"given":"Rui-Qiang","family":"Wang","sequence":"first","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"State Key Laboratory of Cryptology, Beijing 100878, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhen-Qiang","family":"Yin","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"State Key Laboratory of Cryptology, Beijing 100878, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao-Hang","family":"Jin","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"State Key Laboratory of Cryptology, Beijing 100878, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Hong Kong, Pokfulam, Hong Kong"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1822-1613","authenticated-orcid":false,"given":"Shuang","family":"Wang","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"State Key Laboratory of Cryptology, Beijing 100878, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Chen","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"State Key Laboratory of Cryptology, Beijing 100878, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guang-Can","family":"Guo","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"State Key Laboratory of Cryptology, Beijing 100878, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zheng-Fu","family":"Han","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"},{"name":"State Key Laboratory of Cryptology, Beijing 100878, 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