najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМОnajo-7Research ArticlePHYSICSФИЗИКАProperties of multi-moded phase-randomized coherent stateshttps://orcid.org/0000-0002-0809-8431GuselnikovMikhail Sergeevich

Research and Education Center of Photonics and Optical IT, Engineer. Postgraduate student.

msguselnikov@itmo.ruhttps://orcid.org/0000-0001-9870-9285GaidashAndrei Alekseevich

Research and Education Center of Photonics and Optical IT, Lead researcher (ITMO University). Department of Mathematical Methods for Quantum Technologies, Scientific Researcher (Steklov Mathematical Institute of Russian Academy of Sciences). PhD.

andrewdgk@gmail.comhttps://orcid.org/0000-0002-4468-5406KozubovAnton Vladimirovich

Research and Education Center of Photonics and Optical IT, Lead researcher (ITMO University). Department of Mathematical Methods for Quantum Technologies, Scientific Researcher (Steklov Mathematical Institute of Russian Academy of Sciences). PhD.

avkozubov@itmo.ruhttps://orcid.org/0000-0002-4265-8818MiroshnichenkoGeorge Petrovich

Higher School of Engineering and Technology, Professor. Doctor of Science.

gpmirosh@gmail.comУниверситет ИТМОITMO UniversityITMO University, Steklov Mathematical Institute of Russian Academy of SciencesRussian FederationITMO UniversityRussian Federation2025050220261637Copyright © Guselnikov M.S., Gaidash A.A., Kozubov A.V., Miroshnichenko G.P., 20252025Guselnikov M.S., Gaidash A.A., Kozubov A.V., Miroshnichenko G.P.Guselnikov M.S., Gaidash A.A., Kozubov A.V., Miroshnichenko G.P.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/7

Phase-randomized coherent states are widely used in 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 the maximum amount of information accessible to an eavesdropper. These findings indicate that phase randomization can be directly applied to subcarrier wave quantum key distribution systems.

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The authors declare that there are no conflicts of interest present.