[go: up one dir, main page]
Skip to main content

Advertisement

Springer Nature Link
Log in

Continuous variable-based quantum communication in the ocean

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Continuous variable-based quantum cryptography (CV-QKD) is an emerging field in quantum information science, offering unprecedented security for communication protocols by harnessing the principles of quantum mechanics. However, ocean environments pose unique challenges to quantum communication due to their distinct properties and characteristics. This work investigates the impact of turbulence on the transmission of Gaussian light beams used in a continuous variable-based quantum key distribution system for underwater quantum communication. The objective is to quantitatively analyze the induced losses and propose methodologies to mitigate their effects. To achieve this, we adopt the widely accepted ABCD matrix formalism, which provides a comprehensive framework for characterizing the propagation of optical beams through different media. Moreover, a numerical simulation framework is developed to assess the resulting losses and evaluate the performance of the proposed system. The implications of these numerical simulation frameworks for the design and optimization of quantum communication systems for oceanic environments are thoroughly discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Zeng, Z., Fu, S., Zhang, H., Dong, Y., Cheng, J.: A survey of underwater optical wireless communications. IEEE Commun. Surv. Tutor. 19(1), 204–238 (2016)

    Article  MATH  Google Scholar 

  2. Hoeher, P.A., Sticklus, J., Harlakin, A.: Underwater optical wireless communications in swarm robotics: a tutorial. IEEE Commun. Surv. Tutor. 23(4), 2630–2659 (2021)

    Article  MATH  Google Scholar 

  3. Cossu, G., Sturniolo, A., Messa, A., Scaradozzi, D., Ciaramella, E.: Full-fledged 10base-t ethernet underwater optical wireless communication system. IEEE J. Sel. Areas Commun. 36(1), 194–202 (2017)

    Article  Google Scholar 

  4. Bennett, C., Brassard, G. i.: “Proc. ieee int. conf. on computers, systems and signal processing, bangalore, india,” in Proc. of the IEEE Int. Conf. on Computers Systems and Signal Processing Bangalore India, 175, (1984)

  5. Shor, P.W., Preskill, J.: Simple proof of security of the bb84 quantum key distribution protocol. Phys. Rev. Lett. 85(2), 441 (2000)

    Article  ADS  MATH  Google Scholar 

  6. Mayers, D.: Unconditional security in quantum cryptography. J. ACM (JACM) 48(3), 351–406 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Loepp, S., Wootters, W.K.: Protecting information: from classical error correction to quantum cryptography. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  8. Grosshans, F., Grangier, P.: Continuous variable quantum cryptography using coherent states. Phys. Rev. Lett. 88(5), 057902 (2002)

    Article  ADS  MATH  Google Scholar 

  9. Srikara, S., Thapliyal, K., Pathak, A.: Continuous variable b92 quantum key distribution protocol using single photon added and subtracted coherent states. Quantum Inf. Process. 19, 1–16 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pirandola, S., Andersen, U.L., Banchi, L., Berta, M., Bunandar, D., Colbeck, R., Englund, D., Gehring, T., Lupo, C., Ottaviani, C., et al.: Advances in quantum cryptography. Adv. Opt. Photonics 12(4), 1012–1236 (2020)

    Article  ADS  Google Scholar 

  11. Ralph, T.C.: Continuous variable quantum cryptography. Phys. Rev. A 61(1), 010303 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Diamanti, E., Leverrier, A.: Distributing secret keys with quantum continuous variables: principle, security and implementations. Entropy 17(9), 6072–6092 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Laudenbach, F., Pacher, C., Fung, C.-H.F., Poppe, A., Peev, M., Schrenk, B., Hentschel, M., Walther, P., Hübel, H.: Continuous-variable quantum key distribution with gaussian modulation-the theory of practical implementations. Adv. Quantum Technol. 1(1), 1800011 (2018)

    Article  Google Scholar 

  14. Ghalaii, M., Pirandola, S.: Quantum communications in a moderate-to-strong turbulent space. Commun. Phys. 5(1), 38 (2022)

    Article  MATH  Google Scholar 

  15. Heim, B., Peuntinger, C., Killoran, N., Khan, I., Wittmann, C., Marquardt, C., Leuchs, G.: Atmospheric continuous-variable quantum communication. New J. Phys. 16(11), 113018 (2014)

    Article  ADS  MATH  Google Scholar 

  16. Sharma, V., Banerjee, S.: Analysis of atmospheric effects on satellite-based quantum communication: a comparative study. Quantum Inf. Process. 18, 1–24 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Sharma, V., Banerjee, S.: “Analysis of quantum key distribution based satellite communication,” in 2018 9th International Conference on Computing, Communication and Networking Technologies (ICCCNT), 1–5, IEEE, (2018)

  18. Meena, Muskan, R., Banerjee, S.: “Analysing qber and secure key rate under various losses for satellite based free space qkd,” (2023)

  19. Huang, P., He, G., Fang, J., Zeng, G.: Performance improvement of continuous-variable quantum key distribution via photon subtraction. Phys. Rev. A 87(1), 012317 (2013)

    Article  ADS  MATH  Google Scholar 

  20. Ye, W., Zhong, H., Wu, X., Hu, L., Guo, Y.: Continuous-variable measurement-device-independent quantum key distribution via quantum catalysis. Quantum Inf. Process. 19, 1–22 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Li, Z., Zhang, Y., Wang, X., Xu, B., Peng, X., Guo, H.: Non-gaussian postselection and virtual photon subtraction in continuous-variable quantum key distribution. Phys. Rev. A 93(1), 012310 (2016)

    Article  ADS  MATH  Google Scholar 

  22. Giuliano, G.: Underwater optical communication systems. PhD thesis, University of Glasgow, (2019)

  23. Gussen, C.M., Diniz, P.S., Campos, M.L., Martins, W.A., Costa, F.M., Gois, J.N.: A survey of underwater wireless communication technologies. J. Commun. Inf. Syst. 31(1), 242–255 (2016)

    MATH  Google Scholar 

  24. Tian, P., Liu, X., Yi, S., Huang, Y., Zhang, S., Zhou, X., Hu, L., Zheng, L., Liu, R.: High-speed underwater optical wireless communication using a blue gan-based micro-led. Opt. Express 25(2), 1193–1201 (2017)

    Article  ADS  MATH  Google Scholar 

  25. Wang, J., Lu, C., Li, S., Xu, Z.: 100 m/500 mbps underwater optical wireless communication using an nrz-ook modulated 520 nm laser diode. Opt. Express 27(9), 12171–12181 (2019)

    Article  ADS  Google Scholar 

  26. Zhang, L., Wang, Z., Wei, Z., Chen, C., Wei, G., Fu, H., Dong, Y.: Towards a 20 gbps multi-user bubble turbulent noma uowc system with green and blue polarization multiplexing. Opt. Express 28(21), 31796–31807 (2020)

    Article  ADS  MATH  Google Scholar 

  27. Dai, Y., Chen, X., Yang, X., Tong, Z., Du, Z., Lyu, W., Zhang, C., Zhang, H., Zou, H., Cheng, Y., et al.: 200-m/500-mbps underwater wireless optical communication system utilizing a sparse nonlinear equalizer with a variable step size generalized orthogonal matching pursuit. Opt. Express 29(20), 32228–32243 (2021)

    Article  ADS  Google Scholar 

  28. Fei, C., Wang, Y., Du, J., Chen, R., Lv, N., Zhang, G., Tian, J., Hong, X., He, S.: 100-m/3-gbps underwater wireless optical transmission using a wideband photomultiplier tube (pmt). Opt. Express 30(2), 2326–2337 (2022)

    Article  ADS  Google Scholar 

  29. Duntley, S.Q.: Light in the sea. JOSA 53(2), 214–233 (1963)

    Article  ADS  MATH  Google Scholar 

  30. Gilbert, G., Stoner, T., Jernigan, J.: “Underwater experiments on the polarization, coherence, and scattering properties of a pulsed blue-green laser,” in Underwater Photo Optics I, 7, 8–14, SPIE, (1966)

  31. Borah, D.K., Voelz, D.G.: Estimation of laser beam pointing parameters in the presence of atmospheric turbulence. Appl. Opt. 46(23), 6010–6018 (2007)

    Article  ADS  MATH  Google Scholar 

  32. Muskan, A. Dutta., Banerjee, S., Pathak, A.: “Analysis for satellite-based high-dimensional extended B92 and high-dimensional BB84 quantum key distribution,” arXiv preprint[SPACE]arXiv:2311.00309, (2023)

  33. Coronel, J., Elayoubi, K., Al Ahmadi, A., Al Hosani, S., Al Blooshi, A., Al Ameri, R., Cordette, S., Bouchalkha, A., Alameri, J., Matras, G. et al., “Characterization of infrared laser beam through atmospheric optical turbulence in laboratory environment,” in Free-Space Laser Communications XXXV, 12413, pp. 455–467, SPIE, (2023)

  34. Wu, H., van Iersel, M.: “Experimental study on the effects of oam beams propagating through atmospheric turbulence,” in Laser Communication and Propagation through the Atmosphere and Oceans XII, 12691, pp. 132–142, SPIE, (2023)

  35. Tang, S., Zhang, X., Dong, Y.: “Temporal statistics of irradiance in moving turbulent ocean,” in 2013 MTS/IEEE OCEANS-Bergen, 1–4, IEEE, (2013)

  36. Andrews, L.C., Phillips, R.L.: Laser beam Propag. Rand. Med. Second Edition, Laser Beam Propagation Through Random Media (2005)

  37. Zhang, J., Kou, L., Yang, Y., He, F., Duan, Z.: Monte-carlo-based optical wireless underwater channel modeling with oceanic turbulence. Opt. Commun. 475, 126214 (2020)

    Article  Google Scholar 

  38. Enghiyad, N., Sabbagh, A.G.: Impulse response of underwater optical wireless channel in the presence of turbulence, absorption, and scattering employing monte carlo simulation. JOSA A 39(1), 115–126 (2022)

    Article  ADS  MATH  Google Scholar 

  39. Geldard, C. T., Thompson, J., Popoola, W. O.: “Effects of turbulence induced scattering on underwater optical wireless communications,” arXiv preprint[SPACE]arXiv:2008.01152, (2020)

  40. Xu, D., Yue, P., Yi, X., Liu, J.: Improvement of a monte-carlo-simulation-based turbulence-induced attenuation model for an underwater wireless optical communications channel. JOSA A 39(8), 1330–1342 (2022)

    Article  ADS  MATH  Google Scholar 

  41. Ji, X., Yin, H., Liang, Y., Wang, J., et al.: Analysis of aperture averaging effect and communication system performance of wireless optical channels with from weak to strong turbulence in natural turbid water. Opt. Commun. 528, 129018 (2023)

    Article  Google Scholar 

  42. Hou, W., Woods, S., Jarosz, E., Goode, W., Weidemann, A.: Optical turbulence on underwater image degradation in natural environments. Appl. Opt. 51, 2678–2686 (2012)

    Article  ADS  Google Scholar 

  43. Nootz, G., Jarosz, E., Dalgleish, F.R., Hou, W.: Quantification of optical turbulence in the ocean and its effects on beam propagation. Appl. Opt. 55, 8813–8820 (2016)

    Article  ADS  Google Scholar 

  44. Quan, X., Fry, E.S.: Empirical equation for the index of refraction of seawater. Appl. Opt. 34(18), 3477–3480 (1995)

    Article  ADS  MATH  Google Scholar 

  45. Kogelnik, H.: Imaging of optical modes-resonators with internal lenses. Bell Syst. Tech. J. 44(3), 455–494 (1965)

    Article  MATH  Google Scholar 

  46. Kogelnik, H.: On the propagation of gaussian beams of light through lenslike media including those with a loss or gain variation. Appl. Opt. 4, 1562–1569 (1965)

    Article  ADS  MATH  Google Scholar 

  47. Tu, J., Yeoh, G. H., Liu, C., Tao, Y.: Computational fluid dynamics: a practical approach. Elsevier, (2023)

  48. Jaruwatanadilok, S.: Underwater wireless optical communication channel modeling and performance evaluation using vector radiative transfer theory. IEEE J. Sel. Areas Commun. 26(9), 1620–1627 (2008)

    Article  MATH  Google Scholar 

  49. Malpani, P., Alam, N., Thapliyal, K., Pathak, A., Narayanan, V., Banerjee, S.: Lower-and higher-order nonclassical properties of photon added and subtracted displaced fock states. Ann. Phys. 531(2), 1800318 (2019)

    Article  MATH  Google Scholar 

  50. Meena, R., Banerjee, S.: Characterization of quantumness of non-gaussian states under the influence of gaussian channel. Quantum Inf. Process. 22(8), 298 (2023)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Malpani, P., Thapliyal, K., Alam, N., Pathak, A., Narayanan, V., Banerjee, S.: Impact of photon addition and subtraction on nonclassical and phase properties of a displaced fock state. Opt. Commun. 459, 124964 (2020)

    Article  MATH  Google Scholar 

  52. Zhong, H., Guo, Y., Mao, Y., Ye, W., Huang, D.: Virtual zero-photon catalysis for improving continuous-variable quantum key distribution via gaussian post-selection. Sci. Rep. 10(1), 17526 (2020)

    Article  ADS  MATH  Google Scholar 

  53. Xiang, Y., Wang, Y., Ruan, X., Zuo, Z., Guo, Y.: Improving the discretely modulated underwater continuous-variable quantum key distribution with heralded hybrid linear amplifier. Phys. Scr. 96(6), 065103 (2021)

    Article  ADS  MATH  Google Scholar 

  54. McNeil, G.T.: Metrical fundamentals of underwater lens system. Opt. Eng. 16(2), 128–139 (1977)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Griffiths, D. J.: “Introduction to electrodynamics,” (2005)

  56. Leverrier, A., Grangier, P.: Continuous-variable quantum-key-distribution protocols with a non-gaussian modulation. Phys. Rev. A 83(4), 042312 (2011)

    Article  ADS  MATH  Google Scholar 

  57. Peng, Q., Chen, G., Li, X., Liao, Q., Guo, Y.: Performance improvement of underwater continuous-variable quantum key distribution via photon subtraction. Entropy 21(10), 1011 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. Zhao, Y., Zhang, Y., Xu, B., Yu, S., Guo, H.: Continuous-variable measurement-device-independent quantum key distribution with virtual photon subtraction. Phys. Rev. A 97(4), 042328 (2018)

    Article  ADS  MATH  Google Scholar 

  59. Navascués, M., Grosshans, F., Acin, A.: Optimality of gaussian attacks in continuous-variable quantum cryptography. Phys. Rev. Lett. 97(19), 190502 (2006)

    Article  ADS  MATH  Google Scholar 

  60. García-Patrón, R., Cerf, N.J.: Unconditional optimality of gaussian attacks against continuous-variable quantum key distribution. Phys. Rev. Lett. 97(19), 190503 (2006)

    Article  ADS  MATH  Google Scholar 

  61. Wolf, M.M., Giedke, G., Cirac, J.I.: Extremality of gaussian quantum states. Phys. Rev. Lett. 96(8), 080502 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Farwell, N. H., Korotkova, O.: “Multiple phase-screen simulation of oceanic beam propagation,” in Laser communication and propagation through the atmosphere and oceans III, 9224, pp. 374–380, SPIE, (2014)

  63. Johnson, L.J., Green, R.J., Leeson, M.S.: Underwater optical wireless communications: depth dependent variations in attenuation. Appl. Opt. 52, 7867–7873 (2013)

    Article  ADS  MATH  Google Scholar 

  64. Raymont, J. E.: Plankton & productivity in the oceans: Volume 1: Phytoplankton. Elsevier, (2014)

  65. Uitz, J., Claustre, H., Morel, A., Hooker, S. B.: “Vertical distribution of phytoplankton communities in open ocean: An assessment based on surface chlorophyll,” Journal of Geophysical Research: Oceans, 111(C8), (2006)

  66. Hooker, Stanford, B.: “Seawifs technical report series: an overview of seawifs and ocean color,” NASA Technical Memorandum, I, (1992)

  67. Kameda, T., Matsumura, S.: Chlorophyll biomass off sanriku, northwestern pacific, estimated by ocean color and temperature scanner (octs) and a vertical distribution model. J. Oceanogr. 54, 509–516 (1998)

    Article  Google Scholar 

  68. Pontbriand, C., Farr, N., Ware, J., Preisig, J., Popenoe, H.: “Diffuse high-bandwidth optical communications,” in OCEANS 2008, 1–4, IEEE, (2008)

  69. Bricaud, A., Babin, M., Morel, A., Claustre, H.: Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization. J. Geophys. Res. Oceans 100(C7), 13321–13332 (1995)

    Article  ADS  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank CSIR for the fellowship support. SB acknowledges support from Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of Science and Technology (DST), India, Grant No.:DST/ICPS/QuST/Theme-1/2019/6. SB also acknowledges support from the Interdisciplinary Research Platform (IDRP) on Quantum Information and Computation (QIC) at IIT Jodhpur.

Author information

Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology, Jodhpur, 342030, India

    Ramniwas Meena & Subhashish Banerjee

Authors
  1. Ramniwas Meena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Subhashish Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

1. Ramniwas Meena - Conceptualized and designed the research model. - Contributed to the analyse design and methodology. - Drafted the initial manuscript and revised it critically for intellectual content. - Approved the final version of the manuscript. 2. Subhashish Banerjee - Provided guidance throughout the research process. - Assisted in data interpretation, critically revised the manuscript, and provided valuable intellectual input. - Approved the final version of the manuscript. Both authors have read and approved the final manuscript for submission.

Corresponding author

Correspondence to Ramniwas Meena.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest related to this research.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A

1.1 Absorption

Different substances in ocean have specific absorption spectra, meaning they absorb light at different wavelengths. For example, chlorophyll, a pigment found in marine plants and phytoplankton, absorbs light primarily in the blue and red parts of the spectrum while reflecting green light, giving the ocean its characteristic blue color. The absorption properties of ocean have significant implications for marine life, as they determine the available light for photosynthesis and affect the distribution of marine organisms in the water column. They are also essential for understanding the optical properties of the ocean, which are critical for applications such as remote sensing, oceanography, and underwater communication. The full absorption spectrum multiplied by their respective concentrations, such that [63]

$$\begin{aligned} a(\lambda , d) & =a_w(\lambda )+a_f^{0}(\lambda , d) C_f \exp {\left( {-k_f \lambda }\right) } +a_h^{0}(\lambda , d) C_h \exp {\left( {-k_h \lambda }\right) } \nonumber \\ & \quad + a_c^{0}(\lambda ,d) \left( C_c/C\right) ^{0.602}, \end{aligned}$$
(A1)

where \(a_w\) is the pure water absorption coefficient in \(m^{-1}\), \(a_f^{0}\) is the specific absorption coefficient of fulvic acid in \(m^{-1}\), \(a_h^{0}\) is the specific absorption coefficient of humic acid in \(m^{-1}\), \(a_c^{0}\) is the specific absorption coefficient of chlorophyll in \(m^{-1}\), \(C_f\) is the concentration of fulvic acid in \(mg/m^3\), \(C_h\) is the concentration of humic acid in \(mg/m^3\), \(C_c\) is the concentration of chlorophyll-a in \(mg/m^3\), \(k_f\) is the fulvic acid exponential coefficient (\(k_f = 0.0189 m^{-1}\)), and \(k_h\) is the humic acid exponential coefficient (\(k_h = 0.01105 m^{-1}\)).

1.1.1 Scattering

The scattering spectra in ocean are impacted by two main biological factors: scattering caused by pure water and scattering resulting from particulate substances. The latter can be further divided into two categories: small and large particles, each exhibiting distinct statistical distributions and scattering properties. These factors collectively contribute to the overall scattering behavior in the ocean, influencing how light interacts with the water and its constituents. The full form of the equation is given as [63]:

$$\begin{aligned} b(\lambda , d)=b_w(\lambda )+b_s^{0}(\lambda ) C_s(d) +b_l^{0}(\lambda ) C_l(d), \end{aligned}$$
(A2)

where the pure water scattering coefficient is represented as \(b_w\) in \(m^{-1}\), \(b_{s}^{0}\) is the scattering coefficient for small particulate matter in \({m^{2}/{g}}\), the scattering coefficient for large particulate matter is denoted as \({b_{l}^{0}}\) and is also measured in \({m^2}/g\), \({C_s}\) represents the concentration of small particles in \(g/{m^{3}}\), and \({C_l}\) represents the concentration of large particles in \(g/{m^{3}}\).

Chlorophyll-a is a predominant substance found in phytoplankton, a group of microscopic organisms. These photosynthesizing organisms thrive in the photic or euphotic zone of the ocean, where sunlight can penetrate. For efficient photosynthesis, phytoplankton requires an adequate supply of nutrients, which are usually more abundant in coastal areas due to land runoff and upwelling of subsurface water into the photic zone [64]. The spectral dependencies for the scattering coefficients of small and large particulate matter are given by the following equations:

$$\begin{aligned} b_w(\lambda )&=0.005826(400/\lambda )^{4.322}, \end{aligned}$$
(A3)
$$\begin{aligned} b_s^0(\lambda )&=1.1513(400/\lambda )^{1.7},\end{aligned}$$
(A4)
$$\begin{aligned} b_l^0(\lambda )&=0.3411(400/\lambda )^{0.3}. \end{aligned}$$
(A5)

Scattering contributes much less to the overall attenuation coefficient than absorption, through it is much greater in particulate-rich areas.

Table 2 Parameter values for S1–S9 chlorophyll concentration profiles (adapted from [65]). \(C_\mathrm{{chl\text {-}surf}}\): chlorophyll concentration at ocean surface, \(B_0\): background chlorophyll concentration on the surface, S: vertical gradient of concentration, h: chlorophyll above the background level, \(d_\mathrm{{max}}\): depth where chlorophyll has maximum concentration and \(d_{\infty }\): depth where chlorophyll has negligible concentration
Full size table

Monitoring the near-surface chlorophyll concentration in the ocean is essential, and the NASA SeaWiFS (Sea-viewing Wide Field-of-View Sensor) project employs ocean color observation and quantification to determine chlorophyll concentration [66]. Higher chlorophyll concentrations are typically observed along the equator, east-facing coastlines, and in high latitude regions. In the open ocean, typical chlorophyll concentrations range from 0.01 to 4.0 \({\mathrm{mg/m}}^{3}\), while near-shore levels can be as high as 60 \({\mathrm{mg/m}}^{3}\). The chlorophyll profile over a depth d(m) from the surface \(C_c(d)\) can be modeled as a Gaussian curve that includes five numerically determined parameters and has the generic form [67]:

$$\begin{aligned} C_c(d)=B_0+\textit{S}d+\frac{h}{\sigma \sqrt{2 \pi }} \exp {\left[ -\frac{(d-d_{max})^2}{2 \sigma ^2}\right] }, \end{aligned}$$
(A6)

where \(B_0\) is the background chlorophyll concentration on the surface, \(\textit{S}\) is the vertical gradient of concentration, which is always negative due to the slow decrease in chlorophyll concentration with depth, h is the total chlorophyll above the background level and standard deviation of chlorophyll concentration (\(\sigma \)):

$$\begin{aligned} \sigma =\frac{h}{\sqrt{2\pi \left[ C_{chl}(d_{max})-B_0-Sd_{max}\right] }}. \end{aligned}$$
(A7)

Haltrin proposed a simplified one-parameter model for attenuation, which establishes the relationship between the concentrations of different particulates [68]. These particulate concentrations were determined numerically in relation to the chlorophyll concentration and are expressed as follows:

$$\begin{aligned} C_f&= 1.74098 \cdot C_c \cdot \exp (-0.12327 \cdot C_c), \end{aligned}$$
(A8)
$$\begin{aligned} C_h&= 0.19334 \cdot C_c \cdot \exp (-0.12343 \cdot C_c),\end{aligned}$$
(A9)
$$\begin{aligned} C_s&= 0.01739 \cdot C_c \cdot \exp (-0.11631 \cdot C_c),\end{aligned}$$
(A10)
$$\begin{aligned} C_l&= 0.76284 \cdot C_c \cdot \exp (-0.03092 \cdot C_c). \end{aligned}$$
(A11)

Recall that the specific attenuation of chlorophyll also contained a chlorophyll concentration term; so now too has depth dependency:

$$\begin{aligned} a_c^0 (\lambda ,d)=A(\lambda ) C_c(d)^{-B(\lambda )}, \end{aligned}$$
(A12)

where \(A(\lambda )\) and \(B(\lambda )\) represent empirical constants, a full list of which is given in [69].

Appendix B: covariance matrix of m-Photon-subtracted TMSV state

Suppose \(\gamma ^{(m)}_{AB_2}\) represents the covariance matrix of \(\rho ^{(m)}_{AB_2}\), and it has the following formula:

$$\begin{aligned} \gamma ^{(m)}_{AB_2} = \begin{bmatrix} V_A \mathbb {I} & \phi _{AB}\sigma _z \\ \phi _{AB}\sigma _z & V_B \mathbb {I} \end{bmatrix} \end{aligned}$$
(B1)

where \(V_A\) and \(V_B\) represent the variances in modes \(A\) and \(B_2\), respectively. The covariance between the quadratures of modes \(A\) and \(B_2\) is denoted as \(\phi _{AB}\). The matrices \(\mathbb {I}\) and \(\sigma _z\) are defined as diagonal matrices with elements \((1,1)\) and \((1,-1)\), respectively.

Suppose \(x'\), \(p'\) are the heterodyne measurement results of mode A, and x is the homodyne measurement result of mode B. Then,

$$\begin{aligned} V_A=&\langle {\hat{x}}^{2}_A\rangle = 2 \int {x'^{2} P(x',p',x) dx' dp' dx}-1, \nonumber \\ \phi _{AB}=&\langle {\hat{x}}_A{\hat{x}}_B\rangle =\sqrt{2} \int {x'x P(x',p',x) dx' dp' dx}, \nonumber \\ V_B=&\langle {\hat{x}}^{2}_B\rangle = 2 \int {x^{2} P(x',p',x) dx' dp' dx} . \end{aligned}$$
(B2)

where

$$\begin{aligned} P(x',p',x)=WP({x',p'})|\langle x |\sqrt{T}\alpha \rangle |^{2}, \end{aligned}$$
(B3)

the weighting function W and probability \(P(x',p')\) are described in Sect. 4.

After simplifying Eqs. (B2), the resulting expressions are as follows:

$$\begin{aligned} \langle {\hat{x}}_A^2\rangle&= 2\bar{V} - 1, \\ \langle {\hat{x}}_A{\hat{x}}_B\rangle&= 2\sqrt{T}\zeta \bar{V}, \\ \langle {\hat{x}}_B^2\rangle&= 2 T \zeta ^2 \bar{V} + 1, \end{aligned}$$

where \(\bar{V}\) is defined as \(\bar{V}=\int {x'^{2} W\;P({x',p'}) \,dx' \,dp'}\) and further calculations yield:

$$\begin{aligned} \bar{V} = \frac{m + 1}{1 - T\zeta ^2}. \end{aligned}$$
(B4)

Here, \(m\) represents the subtracted photon, \(\zeta \) is the squeezing parameter of the two-mode squeezed vacuum (TMSV) source, and \(T\) is the transmittance of the beam splitter (\(BS_1\)).

Appendix C

From Eq.(4.6), we have the covariance matrix for the final output state,

$$\begin{aligned} V_{out} =\begin{bmatrix} V_A \mathbb {I} & \sqrt{T_c}\phi _{AB}\sigma _z\\ \sqrt{T_c}\phi _{AB}\sigma _z & T_c(V_B+\chi ) \mathbb {I} \end{bmatrix}= \begin{bmatrix} V_1 \mathbb {I} & \phi \sigma _z \\ \phi \sigma _z & V_2 \mathbb {I} \end{bmatrix} \end{aligned}$$
(C1)

where \(\mathbb {I}\) is diag(1, 1), and \(\sigma _z\) is \(diag(1,-1)\), and Alice always uses heterodyne detection. The mutual information between Alice and Bob can be defined as follows:

$$\begin{aligned} I^{Hom}_{A:B}=\frac{1}{2}\log _2\left( \frac{V_a}{V^{Hom}_{a|b}}\right) , \end{aligned}$$
(C2)

where \(V_a=(V_1+1)/2\), \(V_b=V_2\), and

$$\begin{aligned} {V^{Hom}_{a|b}}=V_a - \frac{\phi ^2}{2V_b}=\frac{V_a+1}{2}- \frac{\phi ^2}{2V_b}. \end{aligned}$$
(C3)

If we assume that Eve can purify the entire system, the Holevo quantity for homodyne measurements can be expressed as:

$$\begin{aligned} \chi ^{Hom}_{BE}=S(E)-S(E|B)=S(AB)-S(A|B), \end{aligned}$$

where S(AB) is a function of the symplectic eigenvalues \(\lambda _{1,2}\) of output matrix, which is

$$\begin{aligned} S(AB) = G\left( \frac{{\lambda _1 - 1}}{2}\right) + G\left( \frac{{\lambda _2 - 1}}{2}\right) , \end{aligned}$$
(C4)

where

$$\begin{aligned} G(x)=(x+1)\log _2(x+1)-x\log _2{x}, \end{aligned}$$
(C5)

and

$$\begin{aligned} \lambda _{1,2} = \frac{1}{2} \left( \Delta \pm \sqrt{\Delta - 4D^2}\right) . \end{aligned}$$
(C6)

Here we have used the notations \(\Delta =V^{2}_{1}+V^{2}_{2}-2\phi ^{2}\), and \(D=V_1 V_2-\phi ^2\). Also, \(S(A|B)=G\left( \frac{{\lambda _3 - 1}}{2}\right) \) is a function of the symplectic eigenvalue \(\lambda _3\) of the covariance matrix \(\gamma ^{b}_A\) of the A mode after Bob’s homodyne detection, where \(\lambda _3 = \sqrt{V_1 \left( V_1 - \frac{\phi ^2}{V_2}\right) } \). The covariance matrix \(\gamma ^{b}_A\) can be described as follow [13]:

$$\begin{aligned}\gamma ^{b}_A=\gamma _A-\frac{1}{V_2}\Sigma _c \Pi _{x,p} \Sigma _c, \end{aligned}$$

where \(\gamma _A=V_1 \mathbb {I}\), \(\Sigma _c=V_2 \sigma _z\), \(\Pi _{x}=diag(1,0)\) (in case x is measured) and \(\Pi _{p}=diag(0,1)\) (in case p is measured).

Appendix D: Channel transmission including excess noise

Suppose Alice and Bob share an input covariance matrix, represented as:

$$\begin{aligned} V_\mathrm{{in}} = \begin{pmatrix} V_A \mathbb {I} & \phi _{AB}\sigma _z \\ \phi _{AB}\sigma _z & V_B \mathbb {I} \end{pmatrix} = \begin{pmatrix} v & w \\ w & v \end{pmatrix}. \end{aligned}$$
(D1)

Here, initially \(V_A=V_B\). The channel loss can be modeled using a beam splitter characterized by its transmittance \( T_c \). To incorporate the noise introduced by the beam splitter, an additional noise mode is added to the covariance matrix. This noise mode is represented by a \( 2 \times 2 \) unit matrix and is integrated into the total state through direct summation, resulting in the following total covariance matrix [13]:

$$\begin{aligned} V_\mathrm{{tot}} = V_\mathrm{{in}} \oplus V_\mathrm{{noise}} = \begin{pmatrix} v & w & 0 \\ w & v & 0 \\ 0 & 0 & n \end{pmatrix}, \end{aligned}$$

where \( n = N \mathbb {I} \).

The symplectic representation of the beam splitter matrix is given by:

$$\begin{aligned} \textrm{BS} = \begin{pmatrix} t & r \\ -r & t \end{pmatrix}, \end{aligned}$$

with \( t = \sqrt{T_c}\; \mathbb {I} \) and \( r = \sqrt{1-T_c}\; \mathbb {I} \). To match the dimensions of the covariance matrix \( V_{out} \), a \( 2 \times 2 \) unitary matrix representing the beam splitter’s action on Alice’s mode is appended, yielding:

$$\begin{aligned} \textrm{BS}_{\textrm{total}} = \mathbb {I} \oplus BS = \begin{pmatrix} \mathbb {I} & 0 & 0 \\ 0 & t & r \\ 0 & -r & t \end{pmatrix}. \end{aligned}$$
(D2)

In this configuration, the beam splitter acts on Bob’s mode and the noise mode, while leaving Alice’s mode unchanged. The transformation of a Gaussian state, represented by a covariance matrix \( V \), under the action of a symplectic operator \( S \), follows the rule:

$$\begin{aligned} V_\mathrm{{total}}' = BS_\mathrm{{total}} V_\mathrm{{tot}} BS_\mathrm{{total}}^T. \end{aligned}$$

Substituting the values, this transformation expands to:

$$\begin{aligned} V_\mathrm{{total}}' = \begin{pmatrix} \mathbb {I} & 0 & 0 \\ 0 & t & r \\ 0 & -r & t \end{pmatrix} \begin{pmatrix} v & w & 0 \\ w & v & 0 \\ 0 & 0 & n \end{pmatrix} \begin{pmatrix} \mathbb {I} & 0 & 0 \\ 0 & t & r \\ 0 & -r & t \end{pmatrix}. \end{aligned}$$

This yields the transformed covariance matrix:

$$\begin{aligned} V_{total}' = \begin{pmatrix} v & tw & -rw \\ tw & t^2v + r^2n & -trv + trn \\ -rw & -trv + trn & r^2v + t^2n \end{pmatrix}. \end{aligned}$$

From this result, the variance of the quadrature in Bob’s mode is:

$$\begin{aligned} t^2v + r^2n = T_c V + (1-T_c)N \mathbb {I}. \end{aligned}$$

Defining the noise input as:

$$\begin{aligned} N = 1 + \frac{\epsilon }{1-T_c}, \end{aligned}$$

the covariance matrix takes its final form:

$$\begin{aligned} V_\mathrm{{out}} = \begin{pmatrix} V_A \mathbb {I} & \sqrt{T_c}\phi _{AB} \sigma _z \\ \sqrt{T_c}\phi _{AB} \sigma _z & T_c\left[ V_B + \frac{1-T_c}{T_c} + \epsilon \right] \end{pmatrix}. \end{aligned}$$
(D3)

This is Eq. (4.6).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meena, R., Banerjee, S. Continuous variable-based quantum communication in the ocean. Quantum Inf Process 24, 46 (2025). https://doi.org/10.1007/s11128-025-04664-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-025-04664-2

Keywords