Abstract

Transmittance fluctuations in turbulent atmospheric channels result in quadrature excess noise which limits applicability of continuous-variable quantum communication. Such fluctuations are commonly caused by beam wandering around the receiving aperture. We study the possibility to stabilize the fluctuations by expanding the beam, and test this channel stabilization in regard of continuous-variable entanglement sharing and quantum key distribution. We perform transmittance measurements of a real free-space atmospheric channel for different beam widths and show that the beam expansion reduces the fluctuations of the channel transmittance by the cost of an increased overall loss. We also theoretically study the possibility to share an entangled state or to establish secure quantum key distribution over the turbulent atmospheric channels with varying beam widths. We show the positive effect of channel stabilization by beam expansion on continuous-variable quantum communication as well as the necessity to optimize the method in order to maximize the secret key rate or the amount of shared entanglement. Being autonomous and not requiring adaptive control of the source and detectors based on characterization of beam wandering, the method of beam expansion can be also combined with other methods aiming at stabilizing the fluctuating free-space atmospheric channels.

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

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2018 (3)

R. Baskov and O. Chumak, “Laser-beam scintillations for weak and moderate turbulence,” Phys. Rev. A 97, 043817 (2018).
[Crossref]

P. Papanastasiou, C. Weedbrook, and S. Pirandola, “Continuous-variable quantum key distribution in uniform fast-fading channels,” Phys. Rev. A 97, 032311 (2018).
[Crossref]

D. Vasylyev, W. Vogel, and A. Semenov, “Theory of atmospheric quantum channels based on the law of total probability,” Phys. Rev. A 97, 063852 (2018).
[Crossref]

2017 (4)

H. Kaushal and G. Kaddoum, “Optical communication in space: Challenges and mitigation techniques,” IEEE Commun. Surv. Tutor. 19, 57–96 (2017).
[Crossref]

I. K. Son and S. Mao, “A survey of free space optical networks,” Digital Communications and Networks 3, 67–77 (2017).
[Crossref]

D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

R. Bedington, J. M. Arrazola, and A. Ling, “Progress in satellite quantum key distribution,” npj Quantum Inf. 3, 30 (2017).
[Crossref]

2016 (5)

M. Bohmann, A. A. Semenov, J. Sperling, and W. Vogel, “Gaussian entanglement in the turbulent atmosphere,” Phys. Rev. A 94, 010302 (2016).
[Crossref]

V. C. Usenko and R. Filip, “Trusted noise in continuous-variable quantum key distribution: A threat and a defense,” Entropy 18, 20 (2016).
[Crossref]

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

A. ArockiaBazilRaj and U. Darusalam, “Performance improvement of terrestrial free-space optical communications by mitigating the focal-spot wandering,” J. Mod. Opt. 63, 2339–2347 (2016).
[Crossref]

D. Vasylyev, A. Semenov, and W. Vogel, “Atmospheric quantum channels with weak and strong turbulence,” Phys. Rev. Lett. 117, 090501 (2016).
[Crossref] [PubMed]

2015 (2)

E. Diamanti and A. Leverrier, “Distributing secret keys with quantum continuous variables: Principle, security and implementations,” Entropy 17, 6072–6092 (2015).
[Crossref]

N. Hosseinidehaj and R. Malaney, “Gaussian entanglement distribution via satellite,” Phys. Rev. A 91, 022304 (2015).
[Crossref]

2014 (3)

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in fso communications,” J. Light. Technol. 32, 1323–1328 (2014).
[Crossref]

L. Ruppert, V. C. Usenko, and R. Filip, “Long-distance continuous-variable quantum key distribution with efficient channel estimation,” Phys. Rev. A 90, 062310 (2014).
[Crossref]

2013 (3)

A. Leverrier, R. García-Patrón, R. Renner, and N. J. Cerf, “Security of continuous-variable quantum key distribution against general attacks,” Phys. Rev. Lett. 110, 030502 (2013).
[Crossref] [PubMed]

P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7, 378–381 (2013).
[Crossref]

T. Eberle, V. Händchen, and R. Schnabel, “Stable control of 10 db two-mode squeezed vacuum states of light,” Opt. Express 21, 11546–11553 (2013).
[Crossref] [PubMed]

2012 (4)

D. Y. Vasylyev, A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012).
[Crossref] [PubMed]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
[Crossref]

2011 (1)

P. Jouguet, S. Kunz-Jacques, and A. Leverrier, “Long-distance continuous-variable quantum key distribution with a Gaussian modulation,” Phys. Rev. A 84, 062317 (2011).
[Crossref]

2010 (3)

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photonics Technol. Lett. 22, 1069–1071 (2010).
[Crossref]

H. Guo, B. Luo, Y. Ren, S. Zhao, and A. Dang, “Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius,” Opt. Lett. 35, 1977–1979 (2010).
[Crossref] [PubMed]

A. Leverrier, F. Grosshans, and P. Grangier, “Finite-size analysis of a continuous-variable quantum key distribution,” Phys. Rev. A 81, 062343 (2010).
[Crossref]

2009 (2)

A. A. Semenov and W. Vogel, “Quantum light in the turbulent atmosphere,” Phys. Rev. A 80, 021802 (2009).
[Crossref]

X. Liu, “Free-space optics optimization models for building sway and atmospheric interference using variable wavelength,” IEEE Trans. Commun. 57, 492–498 (2009).
[Crossref]

2008 (2)

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
[Crossref]

S. Pirandola, S. L. Braunstein, and S. Lloyd, “Characterization of collective Gaussian attacks and security of coherent-state quantum cryptography,” Phys. Rev. Lett. 101, 200504 (2008).
[Crossref]

2007 (2)

S. M. Navidpour, M. Uysal, and M. Kavehrad, “Ber performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Commun. 6, 2813 (2007).
[Crossref]

B. Epple and H. Henniger, “Discussion on design aspects for free-space optical communication terminals,” IEEE Commun. Mag. 45, 62 (2007).
[Crossref]

2006 (4)

J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[Crossref] [PubMed]

G. Berman and A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[Crossref]

M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
[Crossref] [PubMed]

R. García-Patrón and N. J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97, 190503 (2006).
[Crossref] [PubMed]

2005 (2)

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 461, 207–235 (2005).
[Crossref]

2004 (1)

E. J. Lee and V. W. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22, 1896–1906 (2004).
[Crossref]

2003 (2)

F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables,” Quantum Inf. Comput. 3, 535–552 (2003).

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

2002 (3)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
[Crossref] [PubMed]

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

2001 (2)

N. J. Cerf, M. Levy, and G. Van Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

A. S. Holevo and R. F. Werner, “Evaluating capacities of bosonic Gaussian channels,” Phys. Rev. A 63, 032312 (2001).
[Crossref]

1999 (1)

T. C. Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A 61, 010303 (1999).
[Crossref]

1994 (1)

G. Baister and P. Gatenby, “Pointing, acquisition and tracking for optical space communications,” Electron. & Commun. Eng. J. 6, 271–280 (1994).
[Crossref]

1990 (1)

Acín, A.

M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
[Crossref] [PubMed]

Andersen, U. L.

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
[Crossref]

J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[Crossref] [PubMed]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser beam scintillation with applications, vol. 99 (SPIE press, 2001).
[Crossref]

ArockiaBazilRaj, A.

A. ArockiaBazilRaj and U. Darusalam, “Performance improvement of terrestrial free-space optical communications by mitigating the focal-spot wandering,” J. Mod. Opt. 63, 2339–2347 (2016).
[Crossref]

Arrazola, J. M.

R. Bedington, J. M. Arrazola, and A. Ling, “Progress in satellite quantum key distribution,” npj Quantum Inf. 3, 30 (2017).
[Crossref]

Baister, G.

G. Baister and P. Gatenby, “Pointing, acquisition and tracking for optical space communications,” Electron. & Commun. Eng. J. 6, 271–280 (1994).
[Crossref]

Baskov, R.

R. Baskov and O. Chumak, “Laser-beam scintillations for weak and moderate turbulence,” Phys. Rev. A 97, 043817 (2018).
[Crossref]

Bayraktar, Ö.

D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

Bedington, R.

R. Bedington, J. M. Arrazola, and A. Ling, “Progress in satellite quantum key distribution,” npj Quantum Inf. 3, 30 (2017).
[Crossref]

Berman, G.

G. Berman and A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[Crossref]

Bohmann, M.

M. Bohmann, A. A. Semenov, J. Sperling, and W. Vogel, “Gaussian entanglement in the turbulent atmosphere,” Phys. Rev. A 94, 010302 (2016).
[Crossref]

Braunstein, S. L.

S. Pirandola, S. L. Braunstein, and S. Lloyd, “Characterization of collective Gaussian attacks and security of coherent-state quantum cryptography,” Phys. Rev. Lett. 101, 200504 (2008).
[Crossref]

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Brouri, R.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

Burris, H. R.

M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
[Crossref]

Cerf, N. J.

A. Leverrier, R. García-Patrón, R. Renner, and N. J. Cerf, “Security of continuous-variable quantum key distribution against general attacks,” Phys. Rev. Lett. 110, 030502 (2013).
[Crossref] [PubMed]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

R. García-Patrón and N. J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97, 190503 (2006).
[Crossref] [PubMed]

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F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
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P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7, 378–381 (2013).
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J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
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R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
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J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
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R. García-Patrón and N. J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97, 190503 (2006).
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M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
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A. Leverrier, F. Grosshans, and P. Grangier, “Finite-size analysis of a continuous-variable quantum key distribution,” Phys. Rev. A 81, 062343 (2010).
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F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
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F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables,” Quantum Inf. Comput. 3, 535–552 (2003).

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
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A. Leverrier, F. Grosshans, and P. Grangier, “Finite-size analysis of a continuous-variable quantum key distribution,” Phys. Rev. A 81, 062343 (2010).
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M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
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F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
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F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables,” Quantum Inf. Comput. 3, 535–552 (2003).

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
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D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
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Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photonics Technol. Lett. 22, 1069–1071 (2010).
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H. Guo, B. Luo, Y. Ren, S. Zhao, and A. Dang, “Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius,” Opt. Lett. 35, 1977–1979 (2010).
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Heersink, J.

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
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J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
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B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
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V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
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B. Epple and H. Henniger, “Discussion on design aspects for free-space optical communication terminals,” IEEE Commun. Mag. 45, 62 (2007).
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M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in fso communications,” J. Light. Technol. 32, 1323–1328 (2014).
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M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
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P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7, 378–381 (2013).
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P. Jouguet, S. Kunz-Jacques, and A. Leverrier, “Long-distance continuous-variable quantum key distribution with a Gaussian modulation,” Phys. Rev. A 84, 062317 (2011).
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S. M. Navidpour, M. Uysal, and M. Kavehrad, “Ber performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Commun. 6, 2813 (2007).
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B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
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B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
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P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7, 378–381 (2013).
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P. Jouguet, S. Kunz-Jacques, and A. Leverrier, “Long-distance continuous-variable quantum key distribution with a Gaussian modulation,” Phys. Rev. A 84, 062317 (2011).
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L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
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R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
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Lee, E. J.

E. J. Lee and V. W. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22, 1896–1906 (2004).
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B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
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V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
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R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
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J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
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E. Diamanti and A. Leverrier, “Distributing secret keys with quantum continuous variables: Principle, security and implementations,” Entropy 17, 6072–6092 (2015).
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A. Leverrier, R. García-Patrón, R. Renner, and N. J. Cerf, “Security of continuous-variable quantum key distribution against general attacks,” Phys. Rev. Lett. 110, 030502 (2013).
[Crossref] [PubMed]

P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7, 378–381 (2013).
[Crossref]

P. Jouguet, S. Kunz-Jacques, and A. Leverrier, “Long-distance continuous-variable quantum key distribution with a Gaussian modulation,” Phys. Rev. A 84, 062317 (2011).
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A. Leverrier, F. Grosshans, and P. Grangier, “Finite-size analysis of a continuous-variable quantum key distribution,” Phys. Rev. A 81, 062343 (2010).
[Crossref]

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N. J. Cerf, M. Levy, and G. Van Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
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D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
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R. Bedington, J. M. Arrazola, and A. Ling, “Progress in satellite quantum key distribution,” npj Quantum Inf. 3, 30 (2017).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
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S. Pirandola, S. L. Braunstein, and S. Lloyd, “Characterization of collective Gaussian attacks and security of coherent-state quantum cryptography,” Phys. Rev. Lett. 101, 200504 (2008).
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J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
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Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photonics Technol. Lett. 22, 1069–1071 (2010).
[Crossref]

H. Guo, B. Luo, Y. Ren, S. Zhao, and A. Dang, “Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius,” Opt. Lett. 35, 1977–1979 (2010).
[Crossref] [PubMed]

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L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
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M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
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N. Hosseinidehaj and R. Malaney, “Gaussian entanglement distribution via satellite,” Phys. Rev. A 91, 022304 (2015).
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D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
[Crossref]

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
[Crossref]

J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[Crossref] [PubMed]

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M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
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M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
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S. M. Navidpour, M. Uysal, and M. Kavehrad, “Ber performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Commun. 6, 2813 (2007).
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P. Papanastasiou, C. Weedbrook, and S. Pirandola, “Continuous-variable quantum key distribution in uniform fast-fading channels,” Phys. Rev. A 97, 032311 (2018).
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B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
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V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
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L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser beam scintillation with applications, vol. 99 (SPIE press, 2001).
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P. Papanastasiou, C. Weedbrook, and S. Pirandola, “Continuous-variable quantum key distribution in uniform fast-fading channels,” Phys. Rev. A 97, 032311 (2018).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
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S. Pirandola, S. L. Braunstein, and S. Lloyd, “Characterization of collective Gaussian attacks and security of coherent-state quantum cryptography,” Phys. Rev. Lett. 101, 200504 (2008).
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M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
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M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in fso communications,” J. Light. Technol. 32, 1323–1328 (2014).
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C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
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M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
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H. Guo, B. Luo, Y. Ren, S. Zhao, and A. Dang, “Influence of beam wander on uplink of ground-to-satellite laser communication and optimization for transmitter beam radius,” Opt. Lett. 35, 1977–1979 (2010).
[Crossref] [PubMed]

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photonics Technol. Lett. 22, 1069–1071 (2010).
[Crossref]

Renner, R.

A. Leverrier, R. García-Patrón, R. Renner, and N. J. Cerf, “Security of continuous-variable quantum key distribution against general attacks,” Phys. Rev. Lett. 110, 030502 (2013).
[Crossref] [PubMed]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Ruppert, L.

L. Ruppert, V. C. Usenko, and R. Filip, “Long-distance continuous-variable quantum key distribution with efficient channel estimation,” Phys. Rev. A 90, 062310 (2014).
[Crossref]

Scharpf, W. J.

M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
[Crossref]

Schnabel, R.

Semenov, A.

D. Vasylyev, W. Vogel, and A. Semenov, “Theory of atmospheric quantum channels based on the law of total probability,” Phys. Rev. A 97, 063852 (2018).
[Crossref]

D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

D. Vasylyev, A. Semenov, and W. Vogel, “Atmospheric quantum channels with weak and strong turbulence,” Phys. Rev. Lett. 117, 090501 (2016).
[Crossref] [PubMed]

D. Y. Vasylyev, A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012).
[Crossref] [PubMed]

Semenov, A. A.

M. Bohmann, A. A. Semenov, J. Sperling, and W. Vogel, “Gaussian entanglement in the turbulent atmosphere,” Phys. Rev. A 94, 010302 (2016).
[Crossref]

A. A. Semenov and W. Vogel, “Quantum light in the turbulent atmosphere,” Phys. Rev. A 80, 021802 (2009).
[Crossref]

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Son, I. K.

I. K. Son and S. Mao, “A survey of free space optical networks,” Digital Communications and Networks 3, 67–77 (2017).
[Crossref]

Sperling, J.

M. Bohmann, A. A. Semenov, J. Sperling, and W. Vogel, “Gaussian entanglement in the turbulent atmosphere,” Phys. Rev. A 94, 010302 (2016).
[Crossref]

Stell, M. F.

M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
[Crossref]

Suite, M. R.

M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
[Crossref]

Tang, X.

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in fso communications,” J. Light. Technol. 32, 1323–1328 (2014).
[Crossref]

Thurn, A.

D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Tualle-Brouri, R.

F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables,” Quantum Inf. Comput. 3, 535–552 (2003).

Usenko, V. C.

V. C. Usenko and R. Filip, “Trusted noise in continuous-variable quantum key distribution: A threat and a defense,” Entropy 18, 20 (2016).
[Crossref]

L. Ruppert, V. C. Usenko, and R. Filip, “Long-distance continuous-variable quantum key distribution with efficient channel estimation,” Phys. Rev. A 90, 062310 (2014).
[Crossref]

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
[Crossref]

I. Derkach, V. C. Usenko, and R. Filip, “Squeezing-enhanced quantum key distribution over atmospheric channels,” arXiv:1809.10167 [quant-ph] (2018).

Uysal, M.

S. M. Navidpour, M. Uysal, and M. Kavehrad, “Ber performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Commun. 6, 2813 (2007).
[Crossref]

Van Assche, G.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

N. J. Cerf, M. Levy, and G. Van Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

Van Loock, P.

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

Vasylyev, D.

D. Vasylyev, W. Vogel, and A. Semenov, “Theory of atmospheric quantum channels based on the law of total probability,” Phys. Rev. A 97, 063852 (2018).
[Crossref]

D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

D. Vasylyev, A. Semenov, and W. Vogel, “Atmospheric quantum channels with weak and strong turbulence,” Phys. Rev. Lett. 117, 090501 (2016).
[Crossref] [PubMed]

Vasylyev, D. Y.

D. Y. Vasylyev, A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012).
[Crossref] [PubMed]

Vidal, G.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

Vilcheck, M. J.

M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
[Crossref]

Vogel, W.

D. Vasylyev, W. Vogel, and A. Semenov, “Theory of atmospheric quantum channels based on the law of total probability,” Phys. Rev. A 97, 063852 (2018).
[Crossref]

D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

M. Bohmann, A. A. Semenov, J. Sperling, and W. Vogel, “Gaussian entanglement in the turbulent atmosphere,” Phys. Rev. A 94, 010302 (2016).
[Crossref]

D. Vasylyev, A. Semenov, and W. Vogel, “Atmospheric quantum channels with weak and strong turbulence,” Phys. Rev. Lett. 117, 090501 (2016).
[Crossref] [PubMed]

D. Y. Vasylyev, A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012).
[Crossref] [PubMed]

A. A. Semenov and W. Vogel, “Quantum light in the turbulent atmosphere,” Phys. Rev. A 80, 021802 (2009).
[Crossref]

Weedbrook, C.

P. Papanastasiou, C. Weedbrook, and S. Pirandola, “Continuous-variable quantum key distribution in uniform fast-fading channels,” Phys. Rev. A 97, 032311 (2018).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Wenger, J.

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables,” Quantum Inf. Comput. 3, 535–552 (2003).

Werner, R. F.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

A. S. Holevo and R. F. Werner, “Evaluating capacities of bosonic Gaussian channels,” Phys. Rev. A 63, 032312 (2001).
[Crossref]

Winter, A.

I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 461, 207–235 (2005).
[Crossref]

Wittmann, C.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
[Crossref]

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Zeng, G.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

Zhao, S.

Appl. Opt. (1)

Digital Communications and Networks (1)

I. K. Son and S. Mao, “A survey of free space optical networks,” Digital Communications and Networks 3, 67–77 (2017).
[Crossref]

Electron. & Commun. Eng. J. (1)

G. Baister and P. Gatenby, “Pointing, acquisition and tracking for optical space communications,” Electron. & Commun. Eng. J. 6, 271–280 (1994).
[Crossref]

Entropy (2)

V. C. Usenko and R. Filip, “Trusted noise in continuous-variable quantum key distribution: A threat and a defense,” Entropy 18, 20 (2016).
[Crossref]

E. Diamanti and A. Leverrier, “Distributing secret keys with quantum continuous variables: Principle, security and implementations,” Entropy 17, 6072–6092 (2015).
[Crossref]

IEEE Commun. Mag. (1)

B. Epple and H. Henniger, “Discussion on design aspects for free-space optical communication terminals,” IEEE Commun. Mag. 45, 62 (2007).
[Crossref]

IEEE Commun. Surv. Tutor. (1)

H. Kaushal and G. Kaddoum, “Optical communication in space: Challenges and mitigation techniques,” IEEE Commun. Surv. Tutor. 19, 57–96 (2017).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

E. J. Lee and V. W. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22, 1896–1906 (2004).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Y. Ren, A. Dang, B. Luo, and H. Guo, “Capacities for long-distance free-space optical links under beam wander effects,” IEEE Photonics Technol. Lett. 22, 1069–1071 (2010).
[Crossref]

IEEE Trans. Commun. (1)

X. Liu, “Free-space optics optimization models for building sway and atmospheric interference using variable wavelength,” IEEE Trans. Commun. 57, 492–498 (2009).
[Crossref]

IEEE Trans. Wirel. Commun. (1)

S. M. Navidpour, M. Uysal, and M. Kavehrad, “Ber performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wirel. Commun. 6, 2813 (2007).
[Crossref]

J. Light. Technol. (1)

M. Hulea, Z. Ghassemlooy, S. Rajbhandari, and X. Tang, “Compensating for optical beam scattering and wandering in fso communications,” J. Light. Technol. 32, 1323–1328 (2014).
[Crossref]

J. Mod. Opt. (1)

A. ArockiaBazilRaj and U. Darusalam, “Performance improvement of terrestrial free-space optical communications by mitigating the focal-spot wandering,” J. Mod. Opt. 63, 2339–2347 (2016).
[Crossref]

Nat. Commun. (1)

L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, “Continuous variable quantum key distribution with modulated entangled states,” Nat. Commun. 3, 1083 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7, 378–381 (2013).
[Crossref]

Nat. Phys. (1)

R. Dong, M. Lassen, J. Heersink, C. Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, “Experimental entanglement distillation of mesoscopic quantum states,” Nat. Phys. 4, 919 (2008).
[Crossref]

Nature (1)

F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421, 238–241 (2003).
[Crossref] [PubMed]

New J. Phys. (2)

V. C. Usenko, B. Heim, C. Peuntinger, C. Wittmann, C. Marquardt, G. Leuchs, and R. Filip, “Entanglement of gaussian states and the applicability to quantum key distribution over fading channels,” New J. Phys. 14, 093048 (2012).
[Crossref]

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, C. Marquardt, and G. Leuchs, “Atmospheric continuous variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

npj Quantum Inf. (1)

R. Bedington, J. M. Arrazola, and A. Ling, “Progress in satellite quantum key distribution,” npj Quantum Inf. 3, 30 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (15)

D. Vasylyev, W. Vogel, and A. Semenov, “Theory of atmospheric quantum channels based on the law of total probability,” Phys. Rev. A 97, 063852 (2018).
[Crossref]

G. Berman and A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[Crossref]

A. A. Semenov and W. Vogel, “Quantum light in the turbulent atmosphere,” Phys. Rev. A 80, 021802 (2009).
[Crossref]

R. Baskov and O. Chumak, “Laser-beam scintillations for weak and moderate turbulence,” Phys. Rev. A 97, 043817 (2018).
[Crossref]

D. Vasylyev, A. Semenov, W. Vogel, K. Günthner, A. Thurn, Ö. Bayraktar, and C. Marquardt, “Free-space quantum links under diverse weather conditions,” Phys. Rev. A 96, 043856 (2017).
[Crossref]

N. Hosseinidehaj and R. Malaney, “Gaussian entanglement distribution via satellite,” Phys. Rev. A 91, 022304 (2015).
[Crossref]

M. Bohmann, A. A. Semenov, J. Sperling, and W. Vogel, “Gaussian entanglement in the turbulent atmosphere,” Phys. Rev. A 94, 010302 (2016).
[Crossref]

P. Papanastasiou, C. Weedbrook, and S. Pirandola, “Continuous-variable quantum key distribution in uniform fast-fading channels,” Phys. Rev. A 97, 032311 (2018).
[Crossref]

N. J. Cerf, M. Levy, and G. Van Assche, “Quantum distribution of gaussian keys using squeezed states,” Phys. Rev. A 63, 052311 (2001).
[Crossref]

T. C. Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A 61, 010303 (1999).
[Crossref]

A. Leverrier, F. Grosshans, and P. Grangier, “Finite-size analysis of a continuous-variable quantum key distribution,” Phys. Rev. A 81, 062343 (2010).
[Crossref]

L. Ruppert, V. C. Usenko, and R. Filip, “Long-distance continuous-variable quantum key distribution with efficient channel estimation,” Phys. Rev. A 90, 062310 (2014).
[Crossref]

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

A. S. Holevo and R. F. Werner, “Evaluating capacities of bosonic Gaussian channels,” Phys. Rev. A 63, 032312 (2001).
[Crossref]

P. Jouguet, S. Kunz-Jacques, and A. Leverrier, “Long-distance continuous-variable quantum key distribution with a Gaussian modulation,” Phys. Rev. A 84, 062317 (2011).
[Crossref]

Phys. Rev. Lett. (8)

M. Navascués, F. Grosshans, and A. Acín, “Optimality of Gaussian attacks in continuous-variable quantum cryptography,” Phys. Rev. Lett. 97, 190502 (2006).
[Crossref] [PubMed]

R. García-Patrón and N. J. Cerf, “Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution,” Phys. Rev. Lett. 97, 190503 (2006).
[Crossref] [PubMed]

S. Pirandola, S. L. Braunstein, and S. Lloyd, “Characterization of collective Gaussian attacks and security of coherent-state quantum cryptography,” Phys. Rev. Lett. 101, 200504 (2008).
[Crossref]

A. Leverrier, R. García-Patrón, R. Renner, and N. J. Cerf, “Security of continuous-variable quantum key distribution against general attacks,” Phys. Rev. Lett. 110, 030502 (2013).
[Crossref] [PubMed]

F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
[Crossref] [PubMed]

J. Heersink, C. Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U. L. Andersen, “Distillation of squeezing from non-gaussian quantum states,” Phys. Rev. Lett. 96, 253601 (2006).
[Crossref] [PubMed]

D. Vasylyev, A. Semenov, and W. Vogel, “Atmospheric quantum channels with weak and strong turbulence,” Phys. Rev. Lett. 117, 090501 (2016).
[Crossref] [PubMed]

D. Y. Vasylyev, A. Semenov, and W. Vogel, “Toward global quantum communication: beam wandering preserves nonclassicality,” Phys. Rev. Lett. 108, 220501 (2012).
[Crossref] [PubMed]

Proc. Royal Soc. A: Math. Phys. Eng. Sci. (1)

I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states,” Proc. Royal Soc. A: Math. Phys. Eng. Sci. 461, 207–235 (2005).
[Crossref]

Quantum Inf. Comput. (1)

F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables,” Quantum Inf. Comput. 3, 535–552 (2003).

Rev. Mod. Phys. (3)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

S. L. Braunstein and P. Van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).
[Crossref]

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621 (2012).
[Crossref]

Sci. Rep. (1)

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6, 19201 (2016).
[Crossref] [PubMed]

Other (3)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser beam scintillation with applications, vol. 99 (SPIE press, 2001).
[Crossref]

I. Derkach, V. C. Usenko, and R. Filip, “Squeezing-enhanced quantum key distribution over atmospheric channels,” arXiv:1809.10167 [quant-ph] (2018).

M. R. Suite, H. R. Burris, C. I. Moore, M. J. Vilcheck, R. Mahon, C. Jackson, M. F. Stell, M. A. Davis, W. S. Rabinovich, W. J. Scharpf, A. E. Reed, and G. C. Gilbreath, “Fast steering mirror implementation for reduction of focal-spot wander in a long-distance free-space optical communication link,” in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds. (SPIE, 2004).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 A schematic view of the experimental set-up. At the sender Alice we use a telescope to expand the beam and adjust its beam width. Subsequently the beam is sent through our 1.6 km free-space link to the receiver Bob. There we use an achromatic lens with a diameter of a = 150 mm and measure the fluctuating transmission using a PIN photodiode detector and an analogue-to-digital converter. To estimate the aperture-to-beam size ratio we use a CCD camera and a screen.
Fig. 2
Fig. 2 Transmittance distribution profiles for different aperture-to-beam size ratios as indicated at the plots.
Fig. 3
Fig. 3 Characteristics of atmospheric fading channel for larger beam expansion characterized by decreasing aperture-to-beam size ratio a/W. (Left): mean values 〈η〉 (lower solid black line) and η (upper dashed grey line) estimated from the analytical Weibull distribution along with the experimental results (squares and circles respectively) and (right): variance Var ( η ) of the square root of transmittance from the experimental characterization of the channel (points) and from the analytical estimates (solid line) versus aperture-to-beam size ratio.
Fig. 4
Fig. 4 Entanglement and secure key rate for larger beam expansion characterized by decreasing aperture-to-beam size ratio a/W. (Left): Logarithmic negativity of an entangled state shared over the fading channel and (Right) Lower bound on the key rate secure against collective attacks in the fading channel, obtained from the analytical fading distribution (lines) along with the experimental results (points) versus aperture-to-beam size ratio. State variance is 7 SNU (solid black line, red circles) or optimized (dashed black line, blue squares), channel excess noise is 1% SNU, post-processing efficiency for the Gaussian CV QKD is 97%.
Fig. 5
Fig. 5 Lower bound on the key rate secure against collective attacks in the fading channel, obtained from the analytical fading distribution, versus aperture-to-beam size ratio at σ b 2 = 0.2 (upper, dotted line), σ b 2 = 0.3 (middle, dashed line), σ b 2 = 0.4 (lower, solid line) upon optimized modulation variance, channel excess noise is 1% SNU, post-processing efficiency is 97%.
Fig. 6
Fig. 6 Entanglement and secure key rate versus aperture-to-beam size ratio a/W at different initial resources. (Left): Logarithmic negativity (LN) of an entangled state shared over a fading channel versus aperture-to-beam size ratio and initial logarithmic negativity LN0 and (Right) Lower bound on the key rate (KR) secure against collective attacks in the fading channel versus aperture-to-beam size ratio and modulated state variance V. Channel excess noise is 1% SNU, post-processing efficiency for the Gaussian CV QKD is 97%.

Equations (2)

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LN = max { 0 , log 2 v } ,
KR = β I AB χ BE ,

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