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Continuous-variable quantum key distribution (CVQKD) is an important application of quantum technology, which enables long-distance communicating parties to establish a string of unconditionally secure keys in an insecure environment. However, in a practical CVQKD system, the finite sampling bandwidth of the analog-to-digital converter (ADC) at the receiver may create inaccurate sampling results, leading to errors in parameter estimation process and leaving a security loophole for eavesdroppers. In order to eliminate the finite sampling bandwidth effect, we propose a peak-compensation-based CVQKD scheme, which estimates the discrepancy between the maximum sampling value and the peak value of each pulse based on the characteristics of Gaussian pulse. The maximum sampling values are compensated by the estimated discrepancy, so that the legitimate parties can obtain correct sampling results. We analyze the influence of the finite sampling bandwidth on the security of the system, expounding the specific steps of peak-compensation, comparing the estimated excess noise before and after peak-compensation, and discussing the security of the system under Gaussian collective attacks. Simulation results show that this scheme can greatly improve the accuracy of pulse peak sampling and remove the finite sampling bandwidth effect. Moreover, the channel parameters estimated by the communicating parties are also corrected by using the compensated values. Compared with the scheme without peak-compensation, this scheme eliminates the limitation of the system repetition to the secret key bit rate, and has longer secure transmission distance and higher secret key bit rate. In addition, compared with other methods of solving the finite sampling bandwidth effect, the proposed scheme can be directly implemented in data processing stage after sampling without any additional devices, and thus increasing no complexity of the system.
[1] Yin J, Li Y H, Liao S K, Yang M, Cao Y, Zhang L, Ren J G, Cai W Q, Liu W Y, Li S L, Shu R, Huang Y M, Deng L, Li L, Zhang Q, Liu N L, Chen Y A, Lu C Y, Wang X B, Xu F H, Wang J Y, Peng C Z, Ekert A K, Pan J W 2020 Nature 582 501
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图 1 CVQKD系统的接收端设备结构图. PBS为偏振分束器, BS为光分束器, PM为相位调制器, PIN为光电二极管, ADC为模数转换器
Figure 1. Structure of receiver’s apparatus of a CVQKD system. PBS, polarization beam splitter; BS, beam splitter; PM, phase modulator; PIN, positive intrinsic-negative; ADC, analog-to-digital converter.
图 2 零差探测器输出脉冲的时域波形, 箭头表示采样位置.
${t_{\rm{s}}}$ 为采样间隔,${U_{\rm{p}}}$ 为脉冲的峰值,${U_{\rm{m}}}$ 为最大测量值,${T_0}$ 为脉冲持续时间Figure 2. Time-domain shape of an output pulse from the balanced homodyne detector.
${t_{\rm{s}}}$ , sampling interval;${U_{\rm{p}}}$ , peak value of the pulse;${U_{\rm{m}}}$ , maximal measurement value;${T_0}$ , duration of each pulse.
图 3 (a)有限采样带宽影响下的高斯脉冲时域采样情况; (b)峰值补偿后的采样值. 其中蓝色线表示脉冲时域波形, 红色圆点代表采样值
Figure 3. (a) Sampling positions of Gaussian pulses effected by finite-sampling bandwidth; (b) sampling values after peak compensation. The blue line represents the time-domain shape of the pulses, and the red dots represent the sampled values.
图 4 不同信道过噪声情况下的估计过噪声随系统重复率的变化. 图中PC表示峰值补偿(peak-compensation, PC)
Figure 4. The estimated excess noise as a function of the system repetition rate under different channel excess noise. PC in the figure represents peak-compensation.
图 5 (a)不同系统重复率下的密钥率随传输距离的变化; (b)不同传输距离下的密钥比特率随系统重复率的变化
Figure 5. (a) The secret key rate as a function of the transmission distance under different system repetition rate; (b) the secret bit rate as a function of the system repetition rate under different transmission distance.
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[1] Yin J, Li Y H, Liao S K, Yang M, Cao Y, Zhang L, Ren J G, Cai W Q, Liu W Y, Li S L, Shu R, Huang Y M, Deng L, Li L, Zhang Q, Liu N L, Chen Y A, Lu C Y, Wang X B, Xu F H, Wang J Y, Peng C Z, Ekert A K, Pan J W 2020 Nature 582 501
Google Scholar
[2] Fang X T, Zeng P, Liu H, Zou M, Wu W J, Tang Y L, Sheng Y J, Xiang Y, Zhang W, Li H, Wang Z, You L, Li M J, Chen H, Chen Y A, Zhang Q, Peng C Z, Ma X, Chen T Y, Pan J W 2020 Nat. Photonics 14 422
Google Scholar
[3] Wang B X, Mao Y Q, Shen L, Zhang L, Lan X B, Ge D W, Gao Y Y, Li J H, Tang Y L, Tang S B, Zhang J, Chen T Y, Pan J W 2020 Opt. Express 28 12558
Google Scholar
[4] Zhang Y, Li Z, Chen Z, Weedbrook C, Zhao Y, Wang X, Huang Y, Xu C, Zhang X, Wang Z, Li M, Zhang X, Zheng Z, Chu B, Gao X, Meng N, Cai W, Wang Z, Wang G, Yu S, Guo H 2019 Quantum Sci. and Technol. 4 035006
Google Scholar
[5] 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
[6] Xu F, Ma X, Zhang Q, Lo H K, Pan J W 2020 Rev. Mod. Phys. 92 025002
Google Scholar
[7] 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
[8] Ralph T C 1999 Phys. Rev. A 61 010303
Google Scholar
[9] Grosshans F, Grangier P 2002 Phys. Rev. Lett. 88 057902
Google Scholar
[10] Grosshans F, Cerf N J, Wenger J, Tualle-Brouri R, Grangier P 2003 Quantum Inf. Comput. 3 535
[11] Weedbrook C, Lance A M, Bowen W P, Symul T, Ralph T C, Lam P K 2004 Phys. Rev. Lett. 93 170504
Google Scholar
[12] Pirandola S, Mancini S, Lloyd S, Braunstein S L 2008 Nat. Phys. 4 726
Google Scholar
[13] Leverrier A, Grangier P 2009 Phys. Rev. Lett. 102 180504
Google Scholar
[14] Weedbrook C, Pirandola S, Ralph T C 2012 Phys. Rev. A 86 022318
Google Scholar
[15] Usenko V C, Grosshans F 2015 Phys. Rev. A 92 062337
Google Scholar
[16] Grosshans F, Cerf N J 2004 Phys. Rev. Lett. 92 047905
Google Scholar
[17] Navascués M, Grosshans F, Acín A 2006 Phys. Rev. Lett. 97 190502
Google Scholar
[18] García-Patrón R, Cerf N J 2006 Phys. Rev. Lett. 97 190503
Google Scholar
[19] Renner R, Cirac J I 2009 Phys. Rev. Lett. 102 110504
Google Scholar
[20] Leverrier A, Grosshans F, Grangier P 2010 Phys. Rev. A 81 062343
Google Scholar
[21] Leverrier A 2015 Phys. Rev. Lett. 114 070501
Google Scholar
[22] Jain N, Anisimova E, Khan I, Makarov V, Marquardt C, Leuchs G 2014 New J. Phys. 16 123030
Google Scholar
[23] Jouguet P, Kunz-Jacques S, Diamanti E 2013 Phys. Rev. A 87 062313
Google Scholar
[24] Ma X C, Sun S H, Jiang M S, Liang L M 2013 Phys. Rev. A 88 022339
Google Scholar
[25] Huang J Z, Weedbrook C, Yin Z Q, Wang S, Li H W, Chen W, Guo G C, Han Z F 2013 Phys. Rev. A 87 062329
Google Scholar
[26] Ma X C, Sun S H, Jiang M S, Liang L M 2013 Phys. Rev. A 87 052309
Google Scholar
[27] Huang J Z, Kunz-Jacques S, Jouguet P, Weedbrook C, Yin Z Q, Wang S, Chen W, Guo G C, Han Z F 2014 Phys. Rev. A 89 032304
Google Scholar
[28] Qin H, Kumar R, Alléaume R 2016 Phys. Rev. A 94 012325
Google Scholar
[29] Qin H, Kumar R, Makarov V, Alléaume R 2018 Phys. Rev. A 98 012312
Google Scholar
[30] Zheng Y, Huang P, Huang A, Peng J, Zeng G 2019 Opt. Express 27 27369
Google Scholar
[31] Liu W, Wang X, Wang N, Du S, Li Y 2017 Phys. Rev. A 96
[32] Wang C, Huang P, Huang D, Lin D, Zeng G 2016 Phys. Rev. A 93 022315
Google Scholar
[33] Li H, Wang C, Huang P, Huang D, Wang T, Zeng G 2016 Opt. Express 24 20481
Google Scholar
[34] Huang D, Lin D, Wang C, Liu W, Fang S, Peng J, Huang P, Zeng G 2015 Opt. Express 23 17511
Google Scholar
[35] Wang C, Huang D, Huang P, Lin D, Peng J, Zeng G 2015 Sci. Rep. 5 14607
Google Scholar
[36] Qi B, Huang L L, Qian L, Lo H K 2007 Phys. Rev. A 76 052323
Google Scholar
[37] Huang P, Huang J, Wang T, Li H, Huang D, Zeng G 2017 Phys. Rev. A 95 052302
Google Scholar
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