najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-3-311-316najo-319Research ArticleNANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICSНАНОСИСТЕМЫ: ФИЗИКА, ХИМИЯ, МАТЕМАТИКАProperties of multi-moded phase-randomized coherent statesСвойства многомодовых фазово-усредненных когерентных состоянийhttps://orcid.org/0000-0002-0809-8431ГусельниковМ. С.GuselnikovM. S.

Mikhail S. Guselnikov

3b Kadetskaya Line, 199034 Saint Petersburg

msguselnikov@itmo.ruhttps://orcid.org/0000-0001-9870-9285ГайдашА. А.GaidashA. A.

Andrei A. Gaidash

3b Kadetskaya Line, 199034 Saint Petersburg

8 Gubkina Street, 119991 Moscow

andrewdgk@gmail.comhttps://orcid.org/0000-0002-4265-8818МирошниченкоГ. П.MiroshnichenkoG. P.

George P. Miroshnichenko

3b Kadetskaya Line, 199034 Saint Petersburg

gpmirosh@gmail.comhttps://orcid.org/0000-0002-4468-5406КозубовА. В.KozubovA. V.

Anton V. Kozubov

3b Kadetskaya Line, 199034 Saint Petersburg

8 Gubkina Street, 119991 Moscow

avkozubov@itmo.ruITMO UniversityRussian FederationITMO University; Steklov Mathematical Institute of Russian Academy of SciencesRussian Federation202529062025163311316Copyright © Guselnikov M.S., Gaidash A.A., Miroshnichenko G.P., Kozubov A.V., 20252025Гусельников М.С., Гайдаш А.А., Мирошниченко Г.П., Козубов А.В.Guselnikov M.S., Gaidash A.A., Miroshnichenko G.P., Kozubov A.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/319

Phase-randomized coherent states are widely used in various applications of quantum optics. They are best known to be the core part of decoy-state quantum key distribution protocols with phase-coding. From the perspective of future development of quantum protocol architecture, it is important to determine whether phase randomization can be applied at an arbitrary stage of an optical scheme without affecting the informational properties of the quantum system. In this paper, using the superoperator formalism, we have shown that phase randomization of a two-mode coherent state commutes with linear optical transformations. This implies that phase randomization can be applied virtually at any point within the optical setup. We further demonstrate that the Holevo bound for such a state coincides with that of regular coherent states, bearing in mind that the Holevo bound quantifies only the maximum amount of information accessible to an eavesdropper. Advantages of phase-randomized coherent states compare to regular ones in particular cases of eavesdropper’s strategies should be considered separately. Also, these findings indicate that phase randomization can be directly applied to a subcarrier wave quantum key distribution type of systems, opening prospects for its future development.

Фазово-усредненные когерентные состояния широко используются в протоколах квантового распределения ключей с ложными состояниями и фазовым кодированием. С точки зрения развития архитектуры квантовых протоколов важно понимать, может ли фазовое усреднение осуществляться на произвольном этапе оптической схемы без влияния на информационные свойства квантовой системы. В настоящей работе с помощью формализма алгебры супероператоров показано, что фазовое усреднение двухмодового когерентного состояния коммутирует с линейными оптическими преобразованиями. Это означает, что фазовое усреднение может осуществляться практически на любом этапе оптической установки. Продемонстрировано, что граница Холево для такого состояния совпадает с границей Холево для обычных когерентных состояний, имея в виду, что граница Холево является количественной мерой максимального объема информации, доступного для злоумышленника. Полученные результаты указывают на то, что фазовое усреднение может быть напрямую применено к системам распределения квантовых ключей на боковых частотах.

когерентные состоянияфазовая рандомизацияфазово-усредненные когерентные состоянияквантовое распределение ключейграница Холевоквантовое распределение ключей на боковых частотахcoherent statesphase randomizationphase-averaged coherent statesquantum key distributionHolevo boundsubcarrier wave quantum key distributionContribution to the work of A. A. Gaidash and A. V. Kozubov was financially supported by Russian Science Foundation (project 20-71-10072) and performed at Steklov Mathematical Institute of Russian Academy of Sciences.ReferencesZhang Y., Wei K., Xu F. Generalized Hong-Ou-Mandel quantum interference with phase-randomized weak coherent states. Physical Review A, 2020, 101(3), P. 033823.Zhang Y., Wei K., Xu F. Generalized Hong-Ou-Mandel quantum interference with phase-randomized weak coherent states. Physical Review A, 2020, 101(3), P. 033823.Bennett C., Bessette F., Brassard G.,Salvail L., and Smolin J. Experimental quantum cryptography. 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The authors declare that there are no conflicts of interest present.