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Measurement basis choice is an essential step in the underwater continuous variable quantum key distribution system based on homodyne detection. However, in practice, finite bandwidth of analog-to-digital converter on the receiver’s side is limited, which can result in defects in the measurement basis choice. That is, the receiver cannot accurately modulate the corresponding phase angle on the phase modulator for measurement basis choice to implement homodyne detection. The imperfect measurement basis choice will introduce extra excess noise, which affects the security of underwater continuous variable quantum key distribution scheme. To solve this problem, we propose an underwater continuous variable quantum key distribution scheme based on imperfect measurement basis choice, and analyze the influence of imperfect measurement basis choice on the performance of underwater continuous variable quantum key distribution system in detail. The research results indicate that the extra excess noise introduced by imperfect measurement basis choice can reduce the secret key rate and maximum transmission distance of the underwater Gaussian modulated quantum key distribution, thus reducing the security of the system. In order to achieve reliable underwater continuous variable quantum key distribution, we quantitatively analyze the extra excess noise introduced by choosing the imperfect measurement basis and obtain its security limit. Besides, we also consider the influence of different seawater depths on the security limit of the proposed scheme, effectively solving the security risks caused by the imperfect measurement basis choice. Furthermore, for the proposed scheme, we consider not only its asymptotic security case but also its composable security case, and the performance curves obtained in the latter are tighter than that achieved in the former. The proposed scheme aims to promote the practical process of underwater continuous variable quantum key distribution system and provide theoretical guidance for accurately evaluating the water channel parameters in underwater communication of global quantum communication networks.
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Keywords:
- imperfect measurement basis choice /
- continuous variable quantum key distribution /
- seawater channel /
- seawater depth
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图 1 基于非理想测量基选择的水下CV-QKD制备-测量方案图. RNG为随机数发生器, AM为振幅调制器, PM为相位调制器, MBC表示测量基选择, $ {T_{\text{s}}} $表示海水信道的透过率, $ {\xi _{\text{s}}} $表示海水信道过噪声
Figure 1. Prepare-and-measure version of underwater continuous variable quantum key distribution scheme based on imperfect measurement basis choice. RNG, random number generator; AM, amplitude modulator; PM, phase modulator; MBC, measurement basis choice; $ {T_{\text{s}}} $, the transmittance of seawater channel; $ {\xi _{\text{s}}} $, the excess noise of seawater channel.
图 2 基于非理想测量基选择的水下CV-QKD纠缠模型原理图(QM为量子存储器)
Figure 2. Schematic diagram of the entanglement-based model of underwater continuous variable quantum key distribution scheme based on imperfect basis choice (QM, quantum memory).
图 3 零差探测器的探测原理图(LO为本振光, PM为相位调制器, BS为分束器, PD1(2)为光电探测器)
Figure 3. Principle of balanced homodyne detector. LO, local oscillator; PM, phase modulator; BS, beam splitter; PD1(2), photodetector.
图 4 所提出方案的渐近密钥率与传输距离在不同参数$ \mu $下的关系
Figure 4. Relationship between the asymptotic secret key rate of the proposed scheme and the transmission distance under different parameters $ \mu $.
图 5 所提出方案的渐近密钥率与协商效率在不同参数$ \mu $下的关系
Figure 5. Relationship between the asymptotic secret key rate of the proposed scheme and the reconciliation efficiency under different parameters $ \mu $.
图 6 所提出方案的密钥率与参数$ \mu $在不同海水深度$ h $下的关系
Figure 6. Relationship between the asymptotic secret key rate of the proposed scheme and the parameter $ \mu $ under different seawater depths $ h $.
图 7 非理想测量基选择情况下所提出的方案与基于BL模型的水下CV-QKD方案性能比较
Figure 7. Performance comparison between the proposed scheme and the underwater CV-QKD scheme based on BL model under imperfect measurement basis choice.
图 8 所提出方案的组合密钥率与用于交换的有效脉冲总数在不同参数$ \mu $下的关系
Figure 8. Relationship between the composable secret key rate of the proposed scheme and the number of exchanged signals under different parameters $ \mu $.
图 9 非理想测量基选择情况下所提出的方案与基于BL模型的水下CV-QKD方案组合密钥率比较
Figure 9. Composable secret key rate comparison between the proposed scheme and the underwater CV-QKD scheme based on BL model under imperfect measurement basis choice.
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[1] Zeng Z, Fu S, Zhang H, Dong Y, Cheng J 2017 IEEE Commun. Surv. Tutorials 19 204
Google Scholar
[2] Hanson F, Radic S 2008 Appl. Opt. 47 277
Google Scholar
[3] Kong M, Wang J, Chen Y, Ali T, Sarwar R, Qiu Y, Wang S, Han J, Xu J 2017 Opt. Express 25 21509
Google Scholar
[4] Wang J, Lu C, Li S, Xu Z 2019 Opt. Express 27 12171
Google Scholar
[5] Xu F, Ma X, Zhang Q, Lo H K, Pan J W 2020 Rev. Mod. Phys. 92 025002
Google Scholar
[6] Pirandola S, Andersen U L, Banchi L, Berta M, Bunandar D, Colbeck R, Englund D, Gehring T, Lupo C, Ottaviani C, Pereira J L, Razavi M, Shaari J S, Tomamichel M, Usenko V C, Vallone G, Villoresi P, Wallden P 2020 Adv. Opt. Photonics 12 1012
Google Scholar
[7] Liu Y, Zhang W J, Jiang C, Chen J P, Zhang C, Pan W X, Ma D, Dong H, Xiong J M, Zhang C J, Li H, Wang R C, Wu J, Chen T Y, You L, Wang X B, Zhang Q, Pan J W 2023 Phys. Rev. Lett. 130 210801
Google Scholar
[8] Li W, Zhang L, Tan H, Lu Y, Liao S K, Huang J, Li H, Wang Z, Mao H K, Yan B, Li Q, Liu Y, Zhang Q, Peng C Z, You L, Xu F, Pan J W 2023 Nat. Photonics 17 416
Google Scholar
[9] Zahidy M, Mikkelsen M T, Müller R, Lio B D, Krehbiel M, Wang Y, Bart N, Wieck A D, Ludwig A, Galili M, Forchhammer S, Lodahl P, Oxenløwe L K, Bacco D, Midolo L 2024 npj Quantum Inf. 10 2
Google Scholar
[10] Zhu H T, Huang Y, Liu H, Zeng P, Zou M, Dai Y, Tang S, Li H, You L, Wang Z, Chen Y A, Ma X, Chen T Y, Pan J W 2023 Phys. Rev. Lett. 130 030801
Google Scholar
[11] Grosshans F, Grangier P 2002 Phys. Rev. Lett. 88 057902
Google Scholar
[12] Laudenbach F, Pacher C, Fung C H F, Poppe A, Peev M, Schrenk B, Hentschel M, Walther P, Hübel H 2018 Adv. Quantum Technol. 1 1800011
Google Scholar
[13] Zhang Y, Bian Y, Li Z, Yu S, Guo H 2024 Appl. Phys. Rev. 11 011318
Google Scholar
[14] 吴晓东, 黄端 2023 72 050303
Google Scholar
Wu X D, Huang D 2023 Acta Phys. Sin. 72 050303
Google Scholar
[15] Wu X D, Wang Y J, Zhong H, Liao Q, Guo Y 2019 Front. Phys. 14 41501
Google Scholar
[16] Weedbrook C, Pirandola S, García-Patrón R, Cerf N J, Ralph T C, Shapiro J H, Lloyd S 2012 Rev. Mod. Phys. 84 621
Google Scholar
[17] Renner R, Cirac J I 2009 Phys. Rev. Lett. 102 110504
Google Scholar
[18] Leverrier A, Grosshans F, Grangier P 2010 Phys. Rev. A 81 062343
Google Scholar
[19] Leverrier A, García-Patrón R, Renner R, Cerf N J 2013 Phys. Rev. Lett. 110 030502
Google Scholar
[20] Leverrier A 2015 Phys. Rev. Lett. 114 070501
Google Scholar
[21] Leverrier A 2017 Phys. Rev. Lett. 118 200501
Google Scholar
[22] Grosshans F, Assche G V, Wenger J, Brouri R, Cerf N J, Grangier P 2003 Nature 421 238
Google Scholar
[23] Jouguet P, Kunz-Jacques S, Leverrier A, Grangier P, Diamanti E 2013 Nat. Photonics 7 378
Google Scholar
[24] Huang D, Lin D, Wang C, Liu W, Fang S, Peng J, Huang P, Zeng G 2015 Opt. Express 23 17511
Google Scholar
[25] Huang D, Huang P, Lin D , Zeng G 2016 Sci. Rep. 6 19201
Google Scholar
[26] Zhang G, Haw J Y, Cai H, Xu F, Assad S M, Fitzsimons J F, Zhou X, Zhang Y, Yu S, Wu J, Ser W, Kwek L C, Liu A Q 2019 Nat. Photonics 13 839
Google Scholar
[27] Zhang Y, Chen Z, Pirandola S, Wang X, Zhou C, Chu B, Zhao Y, Xu B, Yu S, Guo H 2020 Phys. Rev. Lett. 125 010502
Google Scholar
[28] Williams B P, Qi B, Alshowkan M, Evans P G, Peters N A 2024 Phys. Rev. Appl. 21 014056
Google Scholar
[29] Hajomer A A E, Derkach I, Jain N, Chin H M, Andersen U L, Gehring T 2024 Sci. Adv. 10 eadi9474
Google Scholar
[30] Grice W P, Qi B 2019 Phys. Rev. A 100 022339
Google Scholar
[31] 吴晓东, 黄端 2024 73 020304
Google Scholar
Wu X D, Huang D 2024 Acta Phys. Sin. 73 020304
Google Scholar
[32] Zhao W, Shi R, Wu X, Wang F, Ruan X 2023 Opt. Express 31 17003
Google Scholar
[33] Shi P, Zhao S C, Gu Y J, Li W D 2015 J. Opt. Soc. Am. A: 32 349
Google Scholar
[34] Zhao S C, Han X H, Xiao Y, Shen Y, Gu Y J, Li W D 2019 J. Opt. Soc. Am. A: 36 883
Google Scholar
[35] Ji L, Gao J, Yang A L, Feng Z, Lin X F, Li Z G, Jin X M 2017 Opt. Express 25 19795
Google Scholar
[36] Feng Z, Li S, Xu Z 2021 Opt. Express 29 8725
Google Scholar
[37] Zhao S, Li W, Shen Y, Yu Y H, Han X H, Zeng H, Cai M, Qian T, Wang S, Wang Z, Xiao Y, Gu Y 2019 Appl. Opt. 58 3902
Google Scholar
[38] Hu C Q, Yan Z Q, Gao J, Li Z M, Zhou H, Dou J P, Jin X M 2021 Phys. Rev. Appl. 15 024060
Google Scholar
[39] Li D D, Shen Q, Chen W, Li Y, Han X, Yang K X, Xu Y, Lin J, Wang C Z, Yong H L, Liu W Y, Cao Y, Yin J, Liao S K, Ren J G 2019 Opt. Commun. 452 220
Google Scholar
[40] Guo Y, Xie C L, Huang P, Li J W, Zhang L, Huang D, Zeng G H 2018 Phys. Rev. A 97 052326
Google Scholar
[41] Xie C L, Guo Y, Wang Y J, Huang D, Zhang L 2018 Chin. Phys. Lett. 35 090302
Google Scholar
[42] Ruan X, Zhang H, Zhao W, Wang X, Li X, Guo Y 2019 Appl. Sci. 9 4956
Google Scholar
[43] Mao Y, Wu X, Huang W, Liao Q, Deng H, Wang Y, Guo Y 2020 Appl. Sci. 10 5744
Google Scholar
[44] Xiang Y, Wang Y, Ruan X, Zuo Z, Guo Y 2021 Phys. Scr. 96 065103
Google Scholar
[45] Tang X, Chen Z, Zhao Z, Kumar R, Dong Y 2022 Opt. Express 30 32428
Google Scholar
[46] Liu W, Peng J, Qi J, Cao Z, He C 2020 Laser Phys. Lett. 17 055203
Google Scholar
[47] Gilerson A, Zhou J, Hlaing S, Ioannou I, Schalles J, Gross B, Moshary F, Ahmed S 2007 Opt. Express 15 15702
Google Scholar
[48] Gariano J, Djordjevic I B 2019 Opt. Express 27 3055
Google Scholar
[49] Fossier S, Diamanti E, Debuisschert T, Tualle-Brouri R, Grangier P 2009 J. Phys. B: At. Mol. Opt. Phys. 42 114014
Google Scholar
[50] Prieur L, Sathyendranath S 1981 Limnol. Oceanogr. 26 671
Google Scholar
[51] Uitz J, Claustre H, Morel A, Hooker S B 2006 J. Geophys. Res. Oceans. 111 C08005
Google Scholar
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