Abstract

Quantum communication has been successfully implemented in optical fibres and through free-space. Fibre systems, though capable of fast key and low error rates, are impractical in communicating with destinations without an established fibre link. Free-space quantum channels can overcome such limitations and reach long distances with the advent of satellite-to-ground links. However, turbulence, resulting from local fluctuations in refractive index, becomes a major challenge by adding errors and losses. Recently, an interest in investigating the possibility of underwater quantum channels has arisen. Here, we investigate the effect of turbulence on an underwater quantum channel using twisted photons in outdoor conditions. We study the effect of turbulence on transmitted error rates, and compare different quantum cryptographic protocols in an underwater quantum channel, showing the feasibility of high-dimensional encoding schemes. Our work may open the way for secure high-dimensional quantum communication between submersibles, and provides important input for potential submersibles-to-satellite quantum communication.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-Relayed Intercontinental Quantum Network,” Phys. Rev. Lett. 120, 030501 (2018).

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted Photons: New Quantum Perspectives in High Dimensions,” Light. Sci. & Appl. 7, 17146 (2018).

2017 (8)

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006 (2017).

M. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. Boyd, M. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).

L. Ji, J. Gao, A.-L. Yang, Z. Feng, X.-F. Lin, Z.-G. Li, and X.-M. Jin, “Performance analysis of quantum key distribution based on air-water channel,” Opt. Express 25, 19795 (2017).

N. T. Islam, C. C. W. Lim, C. Cahall, J. Kim, and D. J. Gauthier, “Provably secure and high-rate quantum key distribution with time-bin qudits,” Sci. Adv. 3, e1701491 (2017).

J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140 (2017).

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Qi Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-Ground Entanglement-Based Quantum Key Distribution,” Phys. Rev. Lett. 119, 200501 (2017).

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

F. Bouchard, R. Fickler, R. W. Boyd, and E. Karimi, “High-dimensional quantum cloning and applications to quantum hacking,” Sci. Adv. 3, e1601915 (2017).

2016 (3)

R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, “Quantum teleportation across a metropolitan fibre network,” Nat. Photonics 10, 676 (2016).

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications,” Sci. Reports 6, 33306 (2016).

J. Baghdady, K. Miller, K. Morgan, M. Byrd, S. Osler, R. Ragusa, W. Li, B. M. Cochenour, and E. G. Johnson, “Multi-Gigabit/s underwater optical communication link using orbital angular momentum multiplexing,” Opt. Express 24, 9794 (2016).

2015 (2)

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).

Y.-Y. Zhou and X.-J. Zhou, “Performance analysis of quantum key distribution based on air-water channel,” Optoelectronics Lett. 11, 149 (2015).

2014 (2)

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20 (2014).

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305 (2007).

2006 (1)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).

2005 (1)

2003 (1)

Y. C. Liang, D. Kaszlikowski, B.-G. Englert, L. C. Kwek, and C. H. Oh, “Tomographic quantum cryptography,” Phys. Rev. A 68, 022324 (2003).

2002 (1)

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).

2001 (1)

J. Rarity, P. Tapster, and P. Gorman, “Secure free-space key exchange to 1.9 km and beyond,” J. Mod. Opt. 48, 1887 (2001).

2000 (1)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).

1998 (2)

W. Buttler, R. Hughes, P. Kwiat, S. Lamoreaux, G. Luther, G. Morgan, J. Nordholt, C. Peterson, and C. Simmons, “Practical free-space quantum key distribution over 1 km,” Phys. Rev. Lett. 81, 3283 (1998).

D. Bruß, “Optimal eavesdropping in quantum cryptography with six states,” Phys. Rev. Lett. 81, 3018 (1998).

1993 (1)

A. Muller, J. Breguet, and N. Gisin, “Experimental demonstration of quantum cryptography using polarized photons in optical fibre over more than 1 km,” EPL (Europhysics Lett. 23, 383 (1993).

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991).

1984 (1)

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf. on Comput. Syst. Signal Process. Bangalore, India 175, 8 (1984).

1982 (1)

1976 (1)

1963 (1)

1941 (1)

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers,” Doklady Akademiia Nauk SSSR 30, 299–303 (1941).

Aguilar, G. H.

R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, “Quantum teleportation across a metropolitan fibre network,” Nat. Photonics 10, 676 (2016).

Ahmed, N.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications,” Sci. Reports 6, 33306 (2016).

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media, vol. 1 (SPIE Press, 2005).

Arbabi, A.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications,” Sci. Reports 6, 33306 (2016).

Arbabi, E.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications,” Sci. Reports 6, 33306 (2016).

Ashrafi, S.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications,” Sci. Reports 6, 33306 (2016).

Baghdady, J.

Banzer, P.

M. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. Boyd, M. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).

Bechmann-Pasquinucci, H.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf. on Comput. Syst. Signal Process. Bangalore, India 175, 8 (1984).

Blauensteiner, B.

Böhm, H.

Bouchard, F.

Bourennane, M.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).

Boyd, R.

M. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. Boyd, M. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).

Boyd, R. W.

F. Bouchard, R. Fickler, R. W. Boyd, and E. Karimi, “High-dimensional quantum cloning and applications to quantum hacking,” Sci. Adv. 3, e1601915 (2017).

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006 (2017).

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20 (2014).

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf. on Comput. Syst. Signal Process. Bangalore, India 175, 8 (1984).

Breguet, J.

A. Muller, J. Breguet, and N. Gisin, “Experimental demonstration of quantum cryptography using polarized photons in optical fibre over more than 1 km,” EPL (Europhysics Lett. 23, 383 (1993).

Bruß, D.

D. Bruß, “Optimal eavesdropping in quantum cryptography with six states,” Phys. Rev. Lett. 81, 3018 (1998).

Buttler, W.

W. Buttler, R. Hughes, P. Kwiat, S. Lamoreaux, G. Luther, G. Morgan, J. Nordholt, C. Peterson, and C. Simmons, “Practical free-space quantum key distribution over 1 km,” Phys. Rev. Lett. 81, 3283 (1998).

Byrd, M.

Cahall, C.

N. T. Islam, C. C. W. Lim, C. Cahall, J. Kim, and D. J. Gauthier, “Provably secure and high-rate quantum key distribution with time-bin qudits,” Sci. Adv. 3, e1701491 (2017).

Cai, W.-Q.

S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-Relayed Intercontinental Quantum Network,” Phys. Rev. Lett. 120, 030501 (2018).

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

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J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

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J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

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S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-Relayed Intercontinental Quantum Network,” Phys. Rev. Lett. 120, 030501 (2018).

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J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

Zeilinger, A.

S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-Relayed Intercontinental Quantum Network,” Phys. Rev. Lett. 120, 030501 (2018).

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted Photons: New Quantum Perspectives in High Dimensions,” Light. Sci. & Appl. 7, 17146 (2018).

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).

K. Resch, M. Lindenthal, B. Blauensteiner, H. Böhm, A. Fedrizzi, C. Kurtsiefer, A. Poppe, T. Schmitt-Manderbach, M. Taraba, R. Ursin, P. Walther, H. Weier, H. Weinfurter, and A. Zeilinger, “Distributing entanglement and single photons through an intra-city, free-space quantum channel,” Opt. Express 13, 202 (2005).

Zhang, L.

S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P. Wang, Y.-A. Chen, X.-B. Wang, M. Steindorfer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Zeilinger, and J.-W. Pan, “Satellite-Relayed Intercontinental Quantum Network,” Phys. Rev. Lett. 120, 030501 (2018).

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Qi Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-Ground Entanglement-Based Quantum Key Distribution,” Phys. Rev. Lett. 119, 200501 (2017).

J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140 (2017).

Zhang, Qi

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Qi Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-Ground Entanglement-Based Quantum Key Distribution,” Phys. Rev. Lett. 119, 200501 (2017).

Zhang, X.-X.

J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140 (2017).

Zhao, S.-C.

P. Shi, S.-C. Zhao, W.-D. Li, and Y.-J. Gu, “Feasibility of underwater free space quantum key distribution,” arXiv preprint arXiv:1402.4666 (2014).

Zhao, Z.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital angular momentum-based space division multiplexing for high-capacity underwater optical communications,” Sci. Reports 6, 33306 (2016).

Zheng, R.-H.

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

Zhou, Q.

R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, “Quantum teleportation across a metropolitan fibre network,” Nat. Photonics 10, 676 (2016).

Zhou, X.-J.

Y.-Y. Zhou and X.-J. Zhou, “Performance analysis of quantum key distribution based on air-water channel,” Optoelectronics Lett. 11, 149 (2015).

Zhou, Y.-L.

J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140 (2017).

Zhou, Y.-Y.

Y.-Y. Zhou and X.-J. Zhou, “Performance analysis of quantum key distribution based on air-water channel,” Optoelectronics Lett. 11, 149 (2015).

Zhu, Z.-C.

J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140 (2017).

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70 (2017).

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Nat. Photonics (1)

R. Valivarthi, M. G. Puigibert, Q. Zhou, G. H. Aguilar, V. B. Verma, F. Marsili, M. D. Shaw, S. W. Nam, D. Oblak, and W. Tittel, “Quantum teleportation across a metropolitan fibre network,” Nat. Photonics 10, 676 (2016).

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Figures (7)

Fig. 1
Fig. 1 Experimental characterization of underwater turbulence. (a) Calculated coefficients for the lowest ten Zernike polynomials from intensity images of a Gaussian beam after propagation through 3 m of water to characterize the turbulence in one particular set of conditions at the time of measurement. The dominant coefficients correspond to oblique and vertical astigmatism (a4 and a6), followed by tip and tilts effects (a2 and a3). The Zernike decomposition was carried out over a disk with a radius of 2.1 mm, which corresponds to 2w0, where w0 = 1.05 mm is the beam radius at the input of the channel. (b) - (c) Evolution of vortex splitting over a 12 s period for a = 2 and = 3 modes, respectively, sent through 5 m of water. The red, blue, and green lines represent the trajectories of the individual singularities, highlighting their splitting and wandering that occurs due to the turbulence.
Fig. 2
Fig. 2 Experimental setup and state cross-talk measurements. (a) Photon pairs (signal λs = 710 nm, idler λi = 943 nm) are generated via spontaneous parametric downconversion pumped from a periodically poled KTP (ppKTP) crystal by a 405 nm diode laser. A long pass filter (IF) blocks the UV and transmits the photon pairs, which are then split at a dichroic mirror (DM). The idler photon is directly detected by a single photon detector (D2) and acts as a heralding trigger for the information-carrying signal photon. Alice prepares the signal photon into a particular state, for example one from the insets, using SLM-A, then sends it to Bob through the 3 m underwater link. Bob performs a measurement on the received state using SLM-B and a single mode optical fibre connected to D1. Coincidence events between D1 and D2 are recorded. (b) Measured cross-talk matrix between the OAM states ( = −3 to 3) that Alice sends and Bob measures. Higher order states experience more cross-talk as compared to lower order states, seen as off-diagonal detection probabilities.
Fig. 3
Fig. 3 Probability-of-detection matrices for d=2 BB84 and Six-states protocols, and the channel process matrix. (a) Theoretical and experimentally measured probability-of-detection matrices for BB84 (left column) and Six-states (right column) protocols in d = 2. We measured QBERs of Q = 6.57% and Q = 6.35 %, respectively, for these two protocols, corresponding to secret key rates of R = 0.301 and R = 0.395. (b) The six-state protocol is a tomographic protocol and can be used to reconstruct the process tomography matrix; the real and imaginary parts of the theoretical matrix are shown in the top row. The experimentally measured process matrix is shown in the bottom row with a process fidelity of = 0.905 .
Fig. 4
Fig. 4 High-dimensional probability-of-detection matrices. Theoretical (top row) and experimentally measured (bottom row) probability-of-detection matrices for BB84 protocols in d = 3 and d = 4. We measured QBER of Q 3D = 11.73% and Q 4D = 29.77%, respectively. The QBER in d = 3 is below the tolerable error threshold, allowing for the establishment of a secret key rate of R 3 D = 0.307 bits per sifted photon. However, the QBER in d = 4 exceeds the threshold of Q 4Dthreshold = 18.9%.
Fig. 5
Fig. 5 Image of outdoor underwater link in an in-ground swimming pool. The single photon source, on the sender’s side, is enclosed within a black box. The sender’s setup is mounted on an optical breadboard and consists of a microscope objective stage, a spatial light modular, mirrors and a periscope which brings the beam in an air-filled aquarium that is half-way under the water. In order to stabilize the aquariums, they are secured to a ladder that is placed above the underwater link, but is not in direct contact with the optical breadboards. The idler photon is sent from the sender to the receiver through a single mode fiber that is mounted along the ladder. The receiver’s setup is mounted on an optical breadboard and also consists of a periscope, mirrors, a spatial light modular and a microscope objective. Red lines were added to outline the beam’s path.
Fig. 6
Fig. 6 Characterization of underwater turbulence. (a) Experimentally obtained coefficients for an input gaussian ( = 0) mode. Inset i, shows an example of a distorted gaussian beam profile at the output of the channel. The Zernike decomposition was carried over a disk with a radius of 2.1 mm, which corresponds to 2w0, where w0 = 1.05 mm is the beam size at the input of the channel. (b) Experimentally obtained coefficients for an input orbital angular momentum beam ( = 1). Inset ii, shows an example of a distorted donut beam profile at the output of the channel. The Zernike decomposition was carried over a disk with a radius of 2.97 mm, which corresponds to 4 w 0 / 2 , where w 0 / 2 = 0.74 mm corresponds to the radius of maximum intensity of the donut beam at the input of the channel.
Fig. 7
Fig. 7 Sender and receiver setup of the outdoor underwater link. (a) Close-up of the single photon source and the sender’s setup. The blue line represents the pump beam and the red lines represent the signal and idler beams. (b) Close up of the receiver’s aquarium, periscope and setup.

Equations (4)

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Φ ( r , φ ) = j a j Z j ( r , φ ) ,
Z even j ( r , φ ) = n + 1 R n m ( r ) 2 cos ( m φ ) , m 0 ,
Z odd j ( r , φ ) = n + 1 R n m ( r ) 2 sin ( m φ ) , m 0 ,
Z j ( r , φ ) = n + 1 R n 0 ( r ) , m = 0 .

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