Математическая модель квантового генератора случайных чисел на основе флуктуации вакуума

Andrey A. Gaidash; Roman K. Goncharov; Anton V. Kozubov; Pavel V. Yakovlev

doi:10.21638/spbu10.2024.202

Authors

Andrey A. Gaidash University ITMO, 49, Kronversky pr., St. Petersburg, 197101, Russian Federation
Roman K. Goncharov ITMO University, Kronverksky Pr. 49, bldg. A, St. Petersburg, 197101, Russia https://orcid.org/0000-0002-9081-8900
Anton V. Kozubov University ITMO, 49, Kronversky pr., St. Petersburg, 197101, Russian Federation
Pavel V. Yakovlev St. Petersburg State University, 7–9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation https://orcid.org/0000-0002-1997-330X

DOI:

https://doi.org/10.21638/spbu10.2024.202

Abstract

A mathematical model of quantum random number generator based on vacuum fluctuations is considered. A mathematical justification for the “randomness” of the generated sequence is given under the assumption of the truth of the basic postulates of quantum theory and the validity of the Poisson probability distribution for the photon flux. The results of experiments and the obtained estimates of the minimum entropy are presented.

Keywords:

quantum random number generator, minimal entropy estimation, vacuum fluctuations

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References

Литература

Ермаков С. М. Метод Монте-Карло и смежные вопросы. М.: Наука, 1975. 472 c.

Прокаев А. Н. Принцип максимума энтропии в теории поиска // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2023. Т. 19. Вып. 1. C. 27–42. https://doi.org/10.21638/11701/spbu10.2023.103

Herrero-Collantes M., Garcia-Escartin J. C. Quantum random number generators // Rev. Mod. Phys. 2017. Vol. 89. N 2. Art. N 015004. https://doi.org/10.1103/RevModPhys.89.015004

Gabriel C., Wittmann C., Sych D., Dong R., Mauerer W., Andersen U. L., Marquardt C., Leuchs G. A generator for unique quantum random numbers based on vacuum states // Nature Photon. 2010. N 4. P. 711–715. https://doi.org/10.1038/nphoton.2010.197

Shi Y., Chng B., Kurtsiefer C. Random numbers from vacuum fluctuations // Applied Physics Letters. 2016. Vol. 109. N 4. Art. N 041101. P. 1–5 https://doi.org/10.1063/1.4959887

Bruynsteen C., Gehring T., Lupo C., Bauwelinck J., Yin X. 100-Gbit/s integrated quantum random number generator based on vacuum fluctuations // PRX Quantum. 2023. Vol. 4. N 1. Art. N 010330. https://doi.org/10.1103/PRXQuantum.4.010330

Gehring T., Lupo C., Kordts A., Solar N. D., Jain N., Rydberg T., Pedersen T. B., Pirandola S., Andersen U. L. Homodyne-based quantum random number generator at 2.9 Gbps secure against quantum side-information // Nature Communications. 2021. Vol. 12. N 1. Art. N 605. https://doi.org/10.1038/s41467-020-20813-w

Drahi D., Walk N., Hoban M. J., Fedorov A. K., Shakhovoy R., Feimov A., Kurochkin Y., Kolthammer W. S., Nunn J., Barrett J., Walmsley I. A. Certified quantum random numbers from untrusted light // Physical Review X. 2020. Vol. 10. N 4. Art. N 041048. https://doi.org/10.48550/arXiv.1905.09665

Шляйх В. П. Квантовая оптика в фазовом пространстве / пер. с англ; под ред. В. П. Яковлева. М.: Физматгиз, 2005. 760 с.

Collett M. J., Loudon R., Gardiner C. W. Quantum theory of optical homodyne and heterodyne detection // Journal of Modern Optics. 1987. Vol. 34. N 6–7. P. 881–902. https://doi.org/10.1080/09500348714550811

Tomamichel M., Schaffner C., Smith A., Renner R. Leftover hashing against quantum side information // IEEE Transactions on Information Theory. 2011. Vol. 57. N 8. P. 5524–5535. https://doi.org/10.1109/TIT.2011.2158473

Carter J. L., Wegman M. N. Universal classes of hash functions // Journal of Computer and System Sciences. 1979. Vol. 18. Iss. 2. P. 143–154. https://doi.org/10.1016/0022-0000(79)90044-8

Mansour Y., Nisan N., Tiwari P. The computational complexity of universal hashing // Theoretical Computer Science. 1993. Vol. 107. P. 121–133. https://doi.org/10.1016/0304-3975(93)90257-T

Ma X., Xu F., Xu H., Tan X., Qi B., Lo H.-K. Postprocessing for quantum random-number generators: entropy evaluation and randomness extraction // Physical Review A. 2013. Vol. 87. N 6. Art. N 062327. https://doi.org/10.1103/PhysRevA.87.062327

Тихонов В. И., Миронов М. А. Марковские процессы. М.: Сов. радио, 1977. 488 c.

Абрамовиц М., Стиган И. Справочник по специальным функциям с формулами, графиками и таблицами. М.: Наука, 1979. 832 с.

Irwin J. O. The frequency distribution of the difference between two independent variates following the same Poisson distribution // Journal of the Royal Statistical Society. Series A. 1937. Vol. 100. Iss. 3. P. 415–416. https://doi.org/10.1111/j.2397-2335.1937.tb04518.x

Skellam J. G. The frequency distribution of the difference between two Poisson variates belonging to different populations // Journal of the Royal Statistical Society. Series A. 1946. Vol. 109. Iss. 3. P. 290–296. https://doi.org/10.1111/j.2397-2335.1946.tb04670.x

Vogel V., Grabow J. Statistics of difference events in homodyne detection // Physical Review A. 1993. Vol. 47. N 5. P. 4227–4235.

Феллер В. Введение в теорию вероятности и ее приложения. В 2 т. / пер. с англ. Ю. В. Прохорова. М.: Мир, 1984. Т. 2. 738 с.

Haw J. Y., Assad S. M., Lance A. M., Ng N. H. Y., Sharma V., Lam P. K., Symul T. Maximization of extractable randomness in a quantum random-number generator // Physical Review Appl. 2015. Vol. 3. N 5. Art. N 054004. https://link.aps.org/doi/10.1103/PhysRevApplied.3.054004

Бендат Д., Пирсол А. Прикладной анализ случайных данных / пер. с англ. В. Е. Привальского, А. И. Кочубинского; под ред. И. Н. Коваленко. М.: Мир, 1989. 540 с.

References

Ermakov S. M. Metod Monte-Karlo i smezhnye voprosy [Monte-Carlo method and related issues]. Moscow, Nauka Publ., 1975, 472 p. (In Russian)

Prokaev A. N. Printsip maksimuma entropii v teorii poiska [The maximum entropy principle in search theory]. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2023, vol. 19, iss. 1, pp. 27–42. https://doi.org/10.21638/11701/spbu10.2023.103 (In Russian)

Herrero-Collantes M., Garcia-Escartin J. C. Quantum random number generators. Rev. Mod. Phys., 2023, vol. 89, no. 2, art. no. 015004. https://doi.org/10.1103/RevModPhys.89.015004

Gabriel C., Wittmann C., Sych D., Dong R., Mauerer W., Andersen U. L., Marquardt C., Leuchs G. A generator for unique quantum random numbers based on vacuum states. Nature Photon, 2010, no. 4, pp. 711–715. https://doi.org/10.1038/nphoton.2010.197

Shi Y., Chng B., Kurtsiefer C. Random numbers from vacuum fluctuations. Applied Physics Letters, 2016, vol. 109, no. 4, art. no. 041101, pp. 1–5. https://doi.org/10.1063/1.4959887

Bruynsteen C., Gehring T., Lupo C., Bauwelinck J., Yin X. 100-Gbit/s integrated quantum random number generator based on vacuum fluctuations. PRX Quantum, 2023, vol. 4, no. 1, art. no. 010330. https://doi.org/10.1103/PRXQuantum.4.010330

Gehring T., Lupo C., Kordts A., Solar N. D., Jain N., Rydberg T., Pedersen T. B., Pirandola S., Andersen U. L. Homodyne-based quantum random number generator at 2.9 Gbps secure against quantum side-information. Nature Communications, 2021, vol. 12, no. 1, art. no. 605. https://doi.org/10.1038/s41467-020-20813-w

Drahi D., Walk N., Hoban M. J., Fedorov A. K., Shakhovoy R., Feimov A., Kurochkin Y., Kolthammer W. S., Nunn J., Barrett J., Walmsley I. A. Certified quantum random numbers from untrusted light. Physical Review X, 2020, vol. 10, no. 4, art. no. 041048. https://doi.org/10.48550/arXiv.1905.09665

Schleich W. P. Kvatovaya optika v fazovom prostranstve [Quantum optics in phase space]. Moscow, Fizmatgiz Publ., 2005, 760 p. (In Russian)

Collett M. J., Loudon R., Gardiner C. W. Quantum theory of optical homodyne and heterodyne detection. Journal of Modern Optics, 1987, vol. 34, no. 6–7, pp. 881–902. https://doi.org/10.1080/09500348714550811

Tomamichel M., Schaffner C., Smith A., Renner R. Leftover hashing against quantum side information. IEEE Transactions on Information Theory, 2011, vol. 57, no. 8, pp. 5524–5535. https://doi.org/10.1109/TIT.2011.2158473

Carter J. L. , Wegman M. N. Universal classes of hash functions. Journal of Computer and System Sciences, 1979, vol. 18, iss. 2, pp. 143–154. https://doi.org/10.1016/0022-0000(79)90044-8

Mansour Y., Nisan N., Tiwari P. The computational complexity of universal hashing. Theoretical Computer Science, 1993, vol. 107, pp. 121–133. https://doi.org/10.1016/0304-3975(93)90257-T

Ma X., Xu F., Xu H., Tan X., Qi B., Lo H-K. Postprocessing for quantum random-number generators: Entropy evaluation and randomness extraction. Physical Review A, 2013, vol. 87, no. 6, art. no. 062327. https://doi.org/10.1103/PhysRevA.87.062327

Tikhonov V. I., Mironov M. A. Markovskie processy [Markov processes]. Moscow, Sov. Radio Publ., 1977, 488 p. (In Russian)

Abramovitz М., Stegun I. Spravochnik po spetsialnym aeyrwbzv s formulami, graphikami i tablitsami [Handbook of mathematical functions with formulas, graphs, and mathematical tables]. Moscow, Nauka Publ., 1979, 832 p. (In Russian)

Irwin J. O. The frequency distribution of the difference between two independent variates following the same Poisson distribution. Journal of the Royal Statistical Society, Series A, 1937, vol. 100, iss. 3, pp. 415–416. https://doi.org/10.1111/j.2397-2335.1937.tb04518.x

Skellam J. G. The frequency distribution of the difference between two Poisson variates belonging to different populations. Journal of the Royal Statistical Society, Series A, 1946, vol. 109, iss. 3, pp. 296–296. https://doi.org/10.1111/j.2397-2335.1946.tb04670.x

Vogel V., Grabow J. Statistics of difference events in homodyne detection Physical Review A, 1993, vol. 47, no. 5, pp. 4227–4235.

Feller W. Vvedenie v teoriyu veroyatnosti i ee prilozheniya [An introduction to probability theory and its applications]. In 2 vol. Moscow, Mir Publ., 1984, vol. 2, 738 p. (In Russian)

Haw J. Y., Assad S. M., Lance A. M., Ng N. H. Y., Sharma V., Lam P. K., Symul T. Maximization of extractable randomness in a quantum random-number generator. Physical Review Appl., 2015, vol. 3, no. 5, art. no. 054004. https://link.aps.org/doi/10.1103/PhysRevApplied.3.054004

Bendat J. S., Piersol A.G. Prikladnoy analiz sluchainyh dannyh [Random data analysis and measurement procedures]. Moscow, Mir Publ., 1989, 540 p. (In Russian)