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In practical quantum key distribution systems, there inevitably exist errors in the quantum state preparation process due to imperfections in realistic equipment and devices. Those errors would lead to some security loopholes in the quantum key distribution systems. According to the work of Tamaki et al. (
Phys. Rev. A 90 052314 ), here in this work we propose a state preparation error tolerant quantum key distribution protocol through using heralded single-photon sources. In this protocol, we characterize the size of the error in the preparation state of Alice and bring it into the security analysis, thereby avoiding possible security loopholes and improving the security of the system. Moreover, we take the three-intensity decoy-state method for example to introduce the method of constructing the model and estimating the parameters, and carry out corresponding numerical simulations. We make a comparison between the loss tolerant protocol with weak coherent source (WCS) and our present protocol using heralded single-photon source (HSPS). Simulation results show that under the same state preparation error, the key generation rate of the protocol based on WCS is higher than that of protocol based on HSPS at short transmission distances (e.g. less than 150 km). The main reason is that the detection efficiency of the local detector used in the latter scheme is low. However, in the case of long transmission distances (e.g. greater than 200 km), the key generation rate of scheme with WCS drops deeply, while the decline of the key generation rate of the present scheme is much flatter. As a result, the former can no longer generate keys after 211 km, while the latter can transmit a maximum distance of 228 km. Moreover, we also make a comparison between the present scheme and the GLLP protocol with HSPS. The simulation results show that the GLLP protocol with HSPS is very sensitive to the state preparation error and its key generation rate will rapidly decrease with the increase of the state preparation error. On the contrary, our present protocol shows almost no performance degradation under practical state preparation errors. It thus verify the robustness against the state preparation errors of our present work. In addition, in principle, the method can also be combined with the measurement-device-independent quantum key distribution protocol and the twin-field quantum key distribution protocol to further increase the secure communication transmission distance that the present system can reach. Therefore, this work may provide an important reference value for the practical application of long-distance quantum secure communication in the near future.-
Keywords:
- quantum key distribution /
- heralded single photon source /
- loss tolerant protocol /
- state preparation errors
[1] Bennett C H, Brassard G 1984 Proceedings of IEEE International Conference on Computers, System and Signal Processing (Vol. 1 of 3) (Bangalore: IEEE) p175
[2] Shor P W, Preskill J 2000 Phys. Rev. Lett. 85 441
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[3] Mayers D 2001 J. ACM 48 351
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Google Scholar
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Google Scholar
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Google Scholar
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[23] 朱峰, 王琴 2014 光学学报 34 0627002
Zhu F, Wang Q 2014 Acta Opt. Sin. 34 0627002
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Google Scholar
[25] Wang Q, Chen W, Xavier G, et al. 2008 Phys. Rev. Lett. 100 090501
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Google Scholar
[27] Zhou Y H, Yu Z W, Wang X B 2016 Phys. Rev. A 93 042324
Google Scholar
[28] Zhang C H, Zhang C M, Guo G C, Wang Q 2018 Opt. Express 26 4219
Google Scholar
[29] Zhou X Y, Zhang C H, Zhang C M, Wang Q 2017 Phys. Rev. A 96 052337
Google Scholar
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Google Scholar
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图 1 实际系统中的标记单光子源的光路装置模型
Figure 1. Model of optical path device for marking single photon source in experimental system.
图 2 基于不同光源的态制备误差容忍协议密钥生成率对比图
Figure 2. Comparison of the key generation rates of the two different state-preparation-error tolerant protocols using either HSPS or WCS.
图 3 在不同态制备误差下, 对比本协议与GLLP协议[24]的密钥生成率随传输距离变化趋势
Figure 3. Comparison of the key generation rate between the present work and GLLP protocol under different state preparation errors.
表 1 在平均光子数强度为0.4时, WCS和HSPS光源中不同光子数脉冲出现的概率
Table 1. Proportion of different photon number pulses in WCS and HSPS when the average photon intensity is 0.4.
光源类型 光子分布 真空态 单光子 多光子 WCS 0.743218 0.221746 0.035036 HSPS 7.432 × 10–7 0.454683 0.545316 
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表 2 基于HSPS的三强度诱骗态态制备误差容忍QKD协议仿真使用的参数列表
Table 2. Parameter list used in simulation of state-preparation-error tolerant QKD protocol for three strength decoy states based on HSPS.
Bob探测器暗
计数率$ {d_{\rm{B}}} $Bob探测器
效率$ {\eta _{\rm{B}}} $系统纠错
系数$ f $Alice探测器暗
计数率$ {d_{\rm{A}}} $Alice探测器探
测效率$ {\eta _A} $信道损耗
系数$ \alpha $/(dB·km–1)0.5 × 10–6 0.15 1.22 10–6 0.75 0.2
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[1] Bennett C H, Brassard G 1984 Proceedings of IEEE International Conference on Computers, System and Signal Processing (Vol. 1 of 3) (Bangalore: IEEE) p175
[2] Shor P W, Preskill J 2000 Phys. Rev. Lett. 85 441
Google Scholar
[3] Mayers D 2001 J. ACM 48 351
Google Scholar
[4] Lo H K, Chau H F 1999 Science 283 2050
Google Scholar
[5] Brassard G, Lütkenhaus N, Mor T, Sanders B C 2000 Phys. Rev. Lett. 85 1330
Google Scholar
[6] Shannon C E 1949 Bell Syst. Tech. J. 28 656
Google Scholar
[7] Gottesman D, Lo H K, Lütkenhaus N, Preskill J 2004 Quantum Inf. Comput. 4 325
[8] Tamaki K, Curty M, Kato G, Lo H K, Azuma K 2014 Phys. Rev. A 90 052314
Google Scholar
[9] Xu F H, Wei K J, Sajeed S H, Kaiser S, Sun S H, Tang Z Y, Qian L, Makarov V, Lo H K 2015 Phys. Rev. A. 92 032305
Google Scholar
[10] Tang Z Y, Wei K J, Bedroya O, Qian L, Lo H K 2016 Phys. Rev. A. 93 042308
Google Scholar
[11] Wang J P, Liu H W, Ma H Q, Sun S H 2019 Phys. Rev. A 99 032309
Google Scholar
[12] Yin Z Q, Fung C H F, Ma X F, Zhang C M, Li H W, Chen W, Wang S, Guo G C, Han Z F 2013 Phys. Rev. A 88 062322
Google Scholar
[13] Zhou X Y, Zhang C M, Guo G C, Wang Q 2019 IEEE Photonics J. 11 7600207
Google Scholar
[14] Zhou X Y, Ding H J, Zhang C H, Li J, Zhang C M, Wang Q 2020 Opt. Lett. 45 4176
Google Scholar
[15] Pereira M, Curty M, Tamaki K 2019 npj Quantum Inf. 5 62
Google Scholar
[16] Wang X B 2005 Phys. Rev. Lett. 94 230503
Google Scholar
[17] Lo H K, Ma X F, Chen K 2005 Phys. Rev. Lett. 94 230504
Google Scholar
[18] Riedmatten H D, Scarani V, Marcikic I, Acín A, Tittel W, Zbinden H, Gisin N 2004 J. Mod. Opt. 51 1637
Google Scholar
[19] Ljunggren D, Tengner M 2005 Phys. Rev. A 72 062301
Google Scholar
[20] Castelletto S, Degiovanni I P, Schettini V, Migdall A 2005 Opt. Express 13 6709
Google Scholar
[21] Pittmann T B, Jacobs B C, Franson J D 2005 Opt. Commun. 246 545
Google Scholar
[22] Mori S, Söderholm J, Namekata N, Inoue S 2006 Opt. Commun. 264 156
Google Scholar
[23] 朱峰, 王琴 2014 光学学报 34 0627002
Zhu F, Wang Q 2014 Acta Opt. Sin. 34 0627002
[24] Wang Q, Wang X B, Guo G C 2007 Phys. Rev. A 75 012312
Google Scholar
[25] Wang Q, Chen W, Xavier G, et al. 2008 Phys. Rev. Lett. 100 090501
Google Scholar
[26] Lütkenhaus N 2000 Phys. Rev. A 61 052304
Google Scholar
[27] Zhou Y H, Yu Z W, Wang X B 2016 Phys. Rev. A 93 042324
Google Scholar
[28] Zhang C H, Zhang C M, Guo G C, Wang Q 2018 Opt. Express 26 4219
Google Scholar
[29] Zhou X Y, Zhang C H, Zhang C M, Wang Q 2017 Phys. Rev. A 96 052337
Google Scholar
[30] Jiang C, Yu Z W, Hu X L, Wang X B 2021 Phys. Rev. A 103 012402
Google Scholar
[31] Lucamarini M, Yuan Z L, Dynes J F, Shields A J 2018 Nature 557 400
Google Scholar
[32] Ma X F, Zeng P, Zhou H Y 2019 Phys. Rev. X 9 029901
Google Scholar
[33] Wang X B, Yu Z W, Hu X L 2018 Phys. Rev. A. 98 062323
Google Scholar
[34] Cui C H, Yin Z Q, Wang R, et al. 2019 Phys. Rev. Appl. 11 034053
Google Scholar
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