Abstract

We employ non-diffractive Bessel–Gaussian beams to investigate the effect of oceanic turbulence on quantum communication protocols via behaviors of quantum-channel capacity and trace distance, based on the analytical expression of the phase structure function of an orbital-angular-momentum (OAM) beam in underwater wireless optical communication. Our results show that turbulence conditions with a larger inner-scale and outer-scale factors, higher dissipation rate of kinetic energy, lower dissipation rate of the mean-squared temperature, and smaller temperature-salinity contribution ratio are beneficial to quantum communication performance. Moreover, we show that the distribution protocol may be improved by distributing quantum superposition states instead of OAM eigenstates. We believe our work provides the first theoretical exploration of quantum-channel capacity in underwater OAM quantum communication.

© 2020 Optical Society of America

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2020 (2)

S. Tarantino, B. D. Lio, D. Cozzolino, and D. Bacco, “Feasibility of quantum communications in aquatic scenarios,” Optik 216, 164639 (2020).
[Crossref]

F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, “Investigation of underwater quantum channels in a 30 meter flume tank using structured photons,” New J. Phys. 22, 093074 (2020).
[Crossref]

2019 (2)

S. Deng, Y. Zhu, and Y. Zhang, “Received probability of vortex modes carried by localized wave of Bessel-Gaussian amplitude envelope in turbulent seawater,” J. Mar. Sci. Eng. 7, 203–214 (2019).
[Crossref]

S. Zhao, W. Zhang, L. Wang, W. Li, L. Gong, W. Cheng, H. Chen, and J. Gruska, “Propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment,” Sci. Rep. 9, 2025 (2019).
[Crossref]

2018 (1)

2017 (2)

Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express 25, 12203–12215 (2017).
[Crossref]

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

2016 (4)

S. Rana, P. Parashar, and M. Lewenstein, “Trace-distance measure of coherence,” Phys. Rev. A 93, 012110 (2016).
[Crossref]

Y. Zhang, Z. Hu, Q. Yu, and Y. Zhu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26, 095001 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A. 33, 1442–1450 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photon. J. 8, 7901411 (2016).
[Crossref]

2014 (1)

2013 (2)

B. Aaronson, R. L. Franco, G. Compagno, and G. Adesso, “Hierarchy and dynamics of trace distance correlations,” New J. Phys. 15, 093022 (2013).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

2012 (3)

2011 (1)

2010 (2)

H. D. Pires, J. Woudenberg, and M. P. van Exter, “Measurement of the orbital angular momentum spectrum of partially coherent beams,” Opt. Lett. 35, 889–891 (2010).
[Crossref]

E. M. Laine, J. Piilo, and H. P. Breuer, “Witness for initial system-environment correlations in open-system dynamics,” Europhys. Lett. 92, 60010 (2010).
[Crossref]

2009 (1)

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

2006 (1)

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[Crossref]

2005 (2)

L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13, 873–881 (2005).
[Crossref]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref]

2004 (1)

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2004).
[Crossref]

2002 (2)

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
[Crossref]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

2000 (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulence fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).
[Crossref]

1998 (1)

S. A. Holevo, “The capacity of the quantum channel with general signal states,” IEEE Trans. Inf. Theory 44, 269–273 (1998).
[Crossref]

1997 (1)

S. Lloyd, “Capacity of the noisy quantum channel,” Phys. Rev. A 55, 1613–1622 (1997).
[Crossref]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Aaronson, B.

B. Aaronson, R. L. Franco, G. Compagno, and G. Adesso, “Hierarchy and dynamics of trace distance correlations,” New J. Phys. 15, 093022 (2013).
[Crossref]

Adesso, G.

B. Aaronson, R. L. Franco, G. Compagno, and G. Adesso, “Hierarchy and dynamics of trace distance correlations,” New J. Phys. 15, 093022 (2013).
[Crossref]

Agnew, M.

Agrawal, B.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Bacco, D.

S. Tarantino, B. D. Lio, D. Cozzolino, and D. Bacco, “Feasibility of quantum communications in aquatic scenarios,” Optik 216, 164639 (2020).
[Crossref]

Bouchard, F.

F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, “Investigation of underwater quantum channels in a 30 meter flume tank using structured photons,” New J. Phys. 22, 093074 (2020).
[Crossref]

Boyd, R. W.

Breuer, H. P.

E. M. Laine, J. Piilo, and H. P. Breuer, “Witness for initial system-environment correlations in open-system dynamics,” Europhys. Lett. 92, 60010 (2010).
[Crossref]

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

Carrasco, S.

Chen, H.

S. Zhao, W. Zhang, L. Wang, W. Li, L. Gong, W. Cheng, H. Chen, and J. Gruska, “Propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment,” Sci. Rep. 9, 2025 (2019).
[Crossref]

Cheng, M.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photon. J. 8, 7901411 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A. 33, 1442–1450 (2016).
[Crossref]

Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22, 22101–22110 (2014).
[Crossref]

Cheng, W.

S. Zhao, W. Zhang, L. Wang, W. Li, L. Gong, W. Cheng, H. Chen, and J. Gruska, “Propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment,” Sci. Rep. 9, 2025 (2019).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).

Compagno, G.

B. Aaronson, R. L. Franco, G. Compagno, and G. Adesso, “Hierarchy and dynamics of trace distance correlations,” New J. Phys. 15, 093022 (2013).
[Crossref]

Cozzolino, D.

S. Tarantino, B. D. Lio, D. Cozzolino, and D. Bacco, “Feasibility of quantum communications in aquatic scenarios,” Optik 216, 164639 (2020).
[Crossref]

Dan, W.

Davidson, F. M.

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
[Crossref]

Deng, S.

S. Deng, Y. Zhu, and Y. Zhang, “Received probability of vortex modes carried by localized wave of Bessel-Gaussian amplitude envelope in turbulent seawater,” J. Mar. Sci. Eng. 7, 203–214 (2019).
[Crossref]

Djordjevic, I.

I. Djordjevic, Quantum Information Processing and Quantum Error Correction: An Engineering Approach (Academic, 2012)

Djordjevic, I. B.

England, D.

F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, “Investigation of underwater quantum channels in a 30 meter flume tank using structured photons,” New J. Phys. 22, 093074 (2020).
[Crossref]

Farwell, N.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[Crossref]

Forbes, A.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
[Crossref]

Franco, R. L.

B. Aaronson, R. L. Franco, G. Compagno, and G. Adesso, “Hierarchy and dynamics of trace distance correlations,” New J. Phys. 15, 093022 (2013).
[Crossref]

Gao, J.

Gao, X.

Gilchrist, A.

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2004).
[Crossref]

Gong, L.

S. Zhao, W. Zhang, L. Wang, W. Li, L. Gong, W. Cheng, H. Chen, and J. Gruska, “Propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment,” Sci. Rep. 9, 2025 (2019).
[Crossref]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Gruska, J.

S. Zhao, W. Zhang, L. Wang, W. Li, L. Gong, W. Cheng, H. Chen, and J. Gruska, “Propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment,” Sci. Rep. 9, 2025 (2019).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Guo, L.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photon. J. 8, 7901411 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A. 33, 1442–1450 (2016).
[Crossref]

Hernandez-Aranda, R. I.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

Heshami, K.

F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, “Investigation of underwater quantum channels in a 30 meter flume tank using structured photons,” New J. Phys. 22, 093074 (2020).
[Crossref]

Holevo, S. A.

S. A. Holevo, “The capacity of the quantum channel with general signal states,” IEEE Trans. Inf. Theory 44, 269–273 (1998).
[Crossref]

Hu, Z.

Y. Zhang, Z. Hu, Q. Yu, and Y. Zhu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26, 095001 (2016).
[Crossref]

Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22, 22101–22110 (2014).
[Crossref]

Huang, Q.

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A. 33, 1442–1450 (2016).
[Crossref]

Hufnagel, F.

F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, “Investigation of underwater quantum channels in a 30 meter flume tank using structured photons,” New J. Phys. 22, 093074 (2020).
[Crossref]

Ibrahim, A. H.

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Karahroudi, M. K.

Karimi, E.

F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, “Investigation of underwater quantum channels in a 30 meter flume tank using structured photons,” New J. Phys. 22, 093074 (2020).
[Crossref]

Konrad, T.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Korotkova, O.

N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
[Crossref]

Laine, E. M.

E. M. Laine, J. Piilo, and H. P. Breuer, “Witness for initial system-environment correlations in open-system dynamics,” Europhys. Lett. 92, 60010 (2010).
[Crossref]

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

Langford, N. K.

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2004).
[Crossref]

Leach, J.

Lewenstein, M.

S. Rana, P. Parashar, and M. Lewenstein, “Trace-distance measure of coherence,” Phys. Rev. A 93, 012110 (2016).
[Crossref]

Li, J.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photon. J. 8, 7901411 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A. 33, 1442–1450 (2016).
[Crossref]

Li, W.

S. Zhao, W. Zhang, L. Wang, W. Li, L. Gong, W. Cheng, H. Chen, and J. Gruska, “Propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment,” Sci. Rep. 9, 2025 (2019).
[Crossref]

Li, Y.

Lio, B. D.

S. Tarantino, B. D. Lio, D. Cozzolino, and D. Bacco, “Feasibility of quantum communications in aquatic scenarios,” Optik 216, 164639 (2020).
[Crossref]

Lloyd, S.

S. Lloyd, “Capacity of the noisy quantum channel,” Phys. Rev. A 55, 1613–1622 (1997).
[Crossref]

McLaren, M.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
[Crossref]

Mobashery, A.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

Moosavi, S. A.

Mouane, O.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

Ndagano, B.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

Nielsen, M. A.

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2004).
[Crossref]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).

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S. Deng, Y. Zhu, and Y. Zhang, “Received probability of vortex modes carried by localized wave of Bessel-Gaussian amplitude envelope in turbulent seawater,” J. Mar. Sci. Eng. 7, 203–214 (2019).
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Appl. Opt. (1)

Europhys. Lett. (1)

E. M. Laine, J. Piilo, and H. P. Breuer, “Witness for initial system-environment correlations in open-system dynamics,” Europhys. Lett. 92, 60010 (2010).
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IEEE Photon. J. (1)

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photon. J. 8, 7901411 (2016).
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Int. J. Fluid Mech. Res. (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulence fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000).
[Crossref]

J. Mar. Sci. Eng. (1)

S. Deng, Y. Zhu, and Y. Zhang, “Received probability of vortex modes carried by localized wave of Bessel-Gaussian amplitude envelope in turbulent seawater,” J. Mar. Sci. Eng. 7, 203–214 (2019).
[Crossref]

J. Opt. Soc. Am. (1)

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. 19, 1794–1802 (2002).
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J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel-Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A. 33, 1442–1450 (2016).
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Laser Phys. (1)

Y. Zhang, Z. Hu, Q. Yu, and Y. Zhu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26, 095001 (2016).
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Nat. Phys. (1)

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
[Crossref]

New J. Phys. (2)

F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, “Investigation of underwater quantum channels in a 30 meter flume tank using structured photons,” New J. Phys. 22, 093074 (2020).
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B. Aaronson, R. L. Franco, G. Compagno, and G. Adesso, “Hierarchy and dynamics of trace distance correlations,” New J. Phys. 15, 093022 (2013).
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Opt. Commun. (2)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
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N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. 285, 872–875 (2012).
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Opt. Express (4)

Opt. Lett. (2)

Optik (1)

S. Tarantino, B. D. Lio, D. Cozzolino, and D. Bacco, “Feasibility of quantum communications in aquatic scenarios,” Optik 216, 164639 (2020).
[Crossref]

Phys. Rev. A (5)

S. Rana, P. Parashar, and M. Lewenstein, “Trace-distance measure of coherence,” Phys. Rev. A 93, 012110 (2016).
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A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2004).
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B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
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[Crossref]

Phys. Rev. Lett. (3)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref]

Sci. Rep. (1)

S. Zhao, W. Zhang, L. Wang, W. Li, L. Gong, W. Cheng, H. Chen, and J. Gruska, “Propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment,” Sci. Rep. 9, 2025 (2019).
[Crossref]

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

I. Djordjevic, Quantum Information Processing and Quantum Error Correction: An Engineering Approach (Academic, 2012)

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Figures (5)

Fig. 1.
Fig. 1. (a) Quantum-channel capacity and (b) trace distance as functions of propagation distance $z$ with different $L$. The total dimension of Hilbert subspace considered is $(2L + 1)$ with OAM modes $m = - L, \ldots ,0, \ldots ,L$, and the initial capacity ${C_0} = \mathop {\log}\nolimits_2 (2L + 1)$. The quantum-channel capacity $C$ at distance $z$ during the propagation is determined by Eqs. (11) and (12).
Fig. 2.
Fig. 2. (a) Quantum-channel capacity and (b) trace distance as functions of propagation distance $z$ and different contribution ratio of temperature and salinity to refractive index fluctuation $\gamma$ with $L = 1$.
Fig. 3.
Fig. 3. (a) Quantum-channel capacity and (b) trace distance as functions of dissipation rate of mean-squared temperature $\chi$ and different dissipation rate of kinetic energy per unit mass of fluid $\epsilon$ with $L = 1$.
Fig. 4.
Fig. 4. (a) Quantum-channel capacity and (b) trace distance as functions of inner-scale factor $\eta$ with different outer-scale factor $\zeta$ and $L = 1$.
Fig. 5.
Fig. 5. (a) Quantum-channel capacity and (b) trace distance as functions of propagation distance $z$ with different cases. We consider $L = 1$ for cases 1 and 2 and $L = 2$ for cases 3 and 4.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

E l 0 ( r , z , ϕ ) = 2 π exp ( i k ρ 2 ω 0 2 z 2 k r 2 4 ( z R i z ) ) × J l 0 ( z R k ρ r z R i z ) exp ( i l 0 ϕ ) ,
U ( r , ϕ , ϕ , z ) = E l 0 ( r , z , ϕ ) E l 0 ( r , z , ϕ ) × exp [ 1 2 D S ( r , r , z ) ] ,
D S ( r , r , z ) = 2 | r r | 2 ρ c 2 ,
ρ c 2 = 8.659 × 10 8 k 2 ( ϵ η ) 1 / 3 ζ 2 χ z ( 1 2.605 γ 1 + 7.013 γ 2 ) .
U ( r , ϕ , ϕ , z ) = 1 2 π l = | α l 0 l ( z ) | 2 exp [ i l ( ϕ ϕ ) ] ,
0 2 π exp [ i n ϕ + τ cos ( ϕ ϕ ) ] d ϕ = 2 π exp ( i n ϕ ) I n ( τ ) ,
| α l 0 l ( z ) | 2 = 4 exp ( k ρ 2 ω 0 2 z 2 + 2 k r 2 z R 2 2 ( z R 2 + z 2 ) ) | J l 0 ( z R k ρ r z R i z ) | 2 × I l l 0 ( 2 r 2 ρ c 2 ) exp ( 2 r 2 ρ c 2 ) ,
S Δ l = 0 | α l 0 l ( z ) | 2 r d r ,
P l , l 0 = S Δ l = 0 Δ l = S Δ l .
| ψ = m = L L c m | m ,
Ξ ( ρ s ) = M ρ s M ,
C ( ρ s ) = max { p m , ρ m } { S [ Ξ ( ρ s ) ] m p m S [ Ξ ( ρ m ) ] } ,
D ( ρ , σ ) = 1 2 T r | ρ σ | = 1 2 T r [ ( ρ σ ) ( ρ σ ) ] .

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