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The effect of the turbulent motion of ocean on the quantum communication based on the orbital angular momentum in an underwater quantum channel is studied in this work. Based on the power spectrum model of ocean turbulence proposed by Elamassie, the quantitative relationships of different ocean turbulence parameters with the single photon detection probability of orbital angular momentum photons, the channel capacity, the key generation rate, the concurrence of two entangled photons are proposed. The maximum entanglement distance of the orbital angular momentum entangled photon-pairs in the ocean turbulence is further studied by the universal entanglement decay of the concurrence of entangled photon-pairs in the ocean turbulence. The results show that the detection probability of single photon, the channel capacity, the key generation rate, and the concurrence of entangled photon-pairs decrease with the increase of the dissipation rate of turbulent kinetic energy and the decrease of the rate of dissipation of mean-squared temperature. The influence of the temperature and salinity balance parameter of ocean turbulence on the performance of underwater quantum communication are significantly different under the condition of whether the stable stratification of seawater is assumed or not. In the ocean turbulent environment, the increasing of the initial orbital angular momentum quantum number of signal photons can improve the key generation rate of quantum key distribution and the resistance of entangled photons to entanglement decay.
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图 1 单光子探测概率与w的关系
Figure 1. Relationship between single photon detection probability and w
图 2 单光子探测概率与
$ {\chi _T} $ 和ε的关系Figure 2. The relationship between single photon detection probability and
$ {\chi _T} $ and ε
图 3 信道容量随各海洋湍流参数的变化关系
Figure 3. The relationship of channel capacity with the ocean turbulence parameters
图 4 密钥产生率随传输距离的变化关系
Figure 4. The relationship of key generation rate with transmission distance
图 5 共生纠缠度与
$ {\chi _T} $ 和ε的关系Figure 5. The relationship between output state concurrence and
$ {\chi _T} $ and ε
图 6 共生纠缠度与z和w的关系
Figure 6. The relationship between output state concurrence and z and w
图 7 共生纠缠度与
$ {l_0} $ 和$ {r_0} $ 的关系Figure 7. The relationship between output state concurrence and
$ {l_0} $ and$ {r_0} $ 
图 8 共生纠缠度与
$ \xi ({l_0})/{r_0} $ 的关系Figure 8. The relationship between output state concurrence and
$ \xi ({l_0})/{r_0} $ 
图 9 共生纠缠度随各海洋湍流参数的变化关系
Figure 9. The relationship between concurrence and the ocean turbulence parameters
表 1 密钥分发系统的仿真参数值
Table 1. Simulation parameters of key distribution system
Parameter $ {P_d} $ f $ {e_d} $ $ \mu (\upsilon ) $ Value $ 3.0 \times {10^{ - 6}} $ $ 1.16 $ 1.5% $ 0.1 $ 
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[1] Jin X M, Ren J G, Yang B, Yi Z H, Zhou F, Xu X F, Wang S K, Yang D, Hu Y F, Jiang S, Yang T, Yin H, Chen K, Peng C Z, Pan J W 2010 Nat. Photonics 4 376
Google Scholar
[2] 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
[3] 聂敏, 王允, 杨光, 张美玲, 裴昌幸 2016 65 020303
Google Scholar
Nie M, Wang Y, Yang G, Zhang M L, Pei C X 2016 Acta Phys. Sin. 65 020303
Google Scholar
[4] 聂敏, 王超旭, 杨光, 张美玲, 孙爱晶, 裴昌幸 2021 70 030301
Nie M, Wang C X, Yang G, Zhang M L, Sun A J, Pei C X 2021 Acta Phys. Sin. 70 030301
[5] 聂敏, 尚鹏钢, 杨光, 张美玲, 裴昌幸 2014 63 240303
Google Scholar
Nie M, Shang P G, Yang G, Zhang M L, Pei C X 2014 Acta Phys. Sin. 63 240303
Google Scholar
[6] 聂敏, 任杰, 杨光, 张美玲, 裴昌幸 2015 64 150301
Google Scholar
Nie M, Ren J, Yang G, Zhang M L, Pei C X 2015 Acta Phys. Sin. 64 150301
Google Scholar
[7] Zhao S C, Li W D, Shen Y, Yu Y H, Han X H, Zeng H, Cai M Q, Qian T, Wang S, Wang Z M, Xiao Y, Gu Y J 2019 Appl. Opt. 58 3902
Google Scholar
[8] 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
[9] Allen L, Beijersbergen M W, Spreeuw R J C, Woerdman J P 1992 Phys. Rev. A 45 8185
Google Scholar
[10] Molina-Terriza G, Torres J P, Torner L 2002 Phys. Rev. Lett. 88 013601
[11] Gibson G, Courtial J, Padgett M J, Vasnetsov M, Pas'ko V, Barnett S M, Franke-Arnold S 2004 Opt. Express 12 5448
Google Scholar
[12] Barreiro J T, Wei T C, Kwiat P G 2008 Nature Phys. 4 282
Google Scholar
[13] Bechmann-Pasquinucci H, Peres A 2000 Phys. Rev. Lett. 85 3313
Google Scholar
[14] Spedalieri F M 2006 Opt. Commun. 260 340
Google Scholar
[15] Paterson C 2005 Phys. Rev. Lett. 94 153901
Google Scholar
[16] Ibrahim A H, Roux F S, McLaren M, Konrad T, Forbes A 2013 Phys. Rev. A 88 012312
Google Scholar
[17] Bouchard F, Sit A, Hufnagel F, Abbas A, Zhang Y W, Heshami K, Fickler R, Marquardt C, Leuchs G, Boyd R W, Karimi E 2018 Opt. Express 26 22563
Google Scholar
[18] 胡涛, 潘孙翔, 王乐, 赵生妹 2018 量子电子学报 35 499
Hu T, Pan S X, Wang L, Zhang S M Chin. J. Quantum Electron. 35 499 (in Chinese)
[19] Cheng M J, Guo L X, Li J T, Huang Q Q, Cheng Q, Zhang D 2016 Appl. Opt. 55 4642
Google Scholar
[20] Elamassie M, Uysal M, Baykal Y, Abdallah M, Qaraqe K 2017 J. Opt. Soc. Am. A 34 1969
Google Scholar
[21] Andrews L C, Phillips R L 2005 Laser Beam Propagation through Random Media (Bellingham, Washington USA: SPIE Press) pp192−206
[22] 吴彤, 季小玲, 李晓庆, 王欢, 邓宇, 丁洲林 2018 67 224206
Google Scholar
Wu T, Ji X L, Li X Q, Wang H, Deng Y, Ding Z L 2018 Acta Phys. Sin. 67 224206
Google Scholar
[23] Fried D L 1966 J. Opt. Soc. Am. 56 1372
Google Scholar
[24] Alonso J R G, Brun T A 2013 Phys. Rev. A 88 022326
Google Scholar
[25] Lo H K, Curty M, Qi B 2012 Phys. Rev. Lett. 108 130503
Google Scholar
[26] Wang L, Zhao S M, Gong L Y, Cheng W W 2015 Chin. Phys. B 24 120307
Google Scholar
[27] Fickler R, Lapkiewicz R, Plick W N, Krenn M, Schaeff C, Ramelow S, Zeilinger A 2012 Science 338 640
Google Scholar
[28] Leonhard N D, Shatokhin V N, Buchleitner A 2015 Phys. Rev. A 91 012345
Google Scholar
[29] Wootters W K 1998 Phys. Rev. Lett. 80 2245
Google Scholar
[30] Yu T, Eberly J H 2006 Opt. Commun. 264 393
Google Scholar
[31] Gao F, Qin S J, Huang W, Wen Q Y 2019 Sci. China-Phys. Mech. Astron. 62 070301
Google Scholar
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