# Improving Underwater Continuous-Variable Measurement-Device-Independent Quantum Key Distribution via Zero-Photon Catalysis

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## Abstract

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## 1. Introduction

## 2. The ZPC-Based MDI-CVQKD Protocol

## 3. Security Analysis

#### 3.1. Derivation of the Secret Key Rate

#### 3.2. Numerical Simulations

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

QKD | Quantum key Distribution |

DVQKD | Discrete-variable Quantum key Distribution |

CVQKD | Continuous-variable Quantum key Distribution |

MDI | Measurement-device-independent |

TMSV | Two-mode squeezed vacuum |

SPS | Single-photon subtraction |

ZPC | Zero-photon catalysis |

EB | Entanglement-based |

PM | Prepare- and-measure |

Het | Heterodyne detection |

Hom | Homodyne detection |

BS | Beam splitter |

## Appendix A. A: Seawater Channel

#### Appendix A.1. Mixing Effects of Temperature and Salinity

#### Appendix A.2. Effects of Sun Elevation Angle

## References

- Vazirani, U.; Vidick, T. Fully Device Independent Quantum Key Distribution. Commun. ACM
**2019**, 62, 133. [Google Scholar] [CrossRef] [Green Version] - Eriksson, T.; Hirano, T.; Puttnam, B.; Rademacher, G.; Luís, R.; Fujiwara, M.; Namiki, R.; Awaji, Y.; Takeoka, M.; Wada, N.; et al. Wavelength division multiplexing of continuous variable quantum key distribution and 18.3 Tbit/s data channels. Commun. Phys.
**2019**, 2, 1301–1350. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.J.; Mao, Y.Y.; Huang, W.T.; Huang, D.; Guo, Y. Optical frequency comb-based multichannel parallel continuous-variable quantum key distribution. Opt. Express
**2019**, 27, 25314–25329. [Google Scholar] [CrossRef] - Gessner, M.; Pezzè, L.; Smerzi, A. Efficient entanglement criteria for discrete, continuous, and hybrid variables. Phys. Rev. A
**2016**, 94, 020101. [Google Scholar] [CrossRef] [Green Version] - 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. arXiv
**2019**, 1906, 01645. [Google Scholar] [CrossRef] [Green Version] - Ye, W.; Zhong, H.; Liao, Q.; Huang, D.; Hu, L.Y.; Guo, Y. Improvement of self-referenced continuous-variable quantum key distribution with quantum photon catalysis. Opt. Express
**2019**, 27, 17186–17198. [Google Scholar] [CrossRef] [PubMed] - Liao, Q.; Guo, Y.; Huang, D.; Huang, P.; Zeng, G. Long-distance continuous-variable quantum key distribution using non-Gaussian state-discrimination detection. New J. Phys.
**2018**, 20, 023015. [Google Scholar] [CrossRef] - Zhao, W.; Guo, Y.; Zhang, L.; Huang, D. Coherent communications; Phase compensation; Phase estimation; Phase noise; Phase shift; Quantum key distribution. Opt. Express
**2019**, 27, 1838–1853. [Google Scholar] [CrossRef] [PubMed] - Shor, P.; Preskill, J. Simple Proof of Security of the BB84 Quantum Key Distribution Protocol. Phys. Rev. Lett.
**2000**, 85, 441–444. [Google Scholar] [CrossRef] [Green Version] - Braunstein, S.; Pirandola, S. Side-Channel-Free Quantum Key Distribution. Phys. Rev. Lett.
**2012**, 108, 130502. [Google Scholar] [CrossRef] [Green Version] - Lo, H.; Curty, M.; Qi, B. Measurement-Device-Independent Quantum Key Distribution. Phys. Rev. Lett.
**2012**, 108, 130503. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pirandola, S.; Ottaviani, C.; Spedalieri, G.; Weedbrook, C.; Braunstein, S.; Lloyd, S.; Gehring, T.; Jacobsen, C.; Andersen, U. High-rate measurement-device-independent quantum cryptography. Nat. Photonics
**2015**, 9, 397–402. [Google Scholar] [CrossRef] [Green Version] - Ma, X.; Sun, S.; Jiang, M.; Liang, L. Local oscillator fluctuation opens a loophole for Eve in practical continuous-variable quantum-key-distribution systems. Phys. Rev. A
**2013**, 88, 022339. [Google Scholar] [CrossRef] [Green Version] - Huang, J.; Weedbrook, C.; Yin, Z.; Wang, S.; Li, H.; Chen, W.; Guo, G.; Han, Z. Quantum hacking of a continuous-variable quantum-key-distribution system using a wavelength attack. Phys. Rev. A
**2013**, 87, 062329. [Google Scholar] [CrossRef] [Green Version] - Qin, H.; Kumar, R.; Alléaume, R. Quantum hacking: Saturation attack on practical continuous-variable quantum key distribution. Phys. Rev. A
**2016**, 94, 012325. [Google Scholar] [CrossRef] [Green Version] - Paterson, C. Atmospheric Turbulence and Orbital Angular Momentum of Single Photons for Optical Communication. Phys. Rev. Lett.
**2005**, 94, 153901. [Google Scholar] [CrossRef] - Berman, G.; Chumak, A. Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere. Phys. Rev. A
**2006**, 74, 013805. [Google Scholar] [CrossRef] [Green Version] - Semenov, A.; Töppel, F.; Vasylyev, D.; Gomonay, H.; Vogel, W. Homodyne detection for atmosphere channels. Phys. Rev. A
**2012**, 85, 013826. [Google Scholar] [CrossRef] [Green Version] - Gariano, J.; Djordjevic, I. Theoretical study of a submarine to submarine quantum key distribution systems. Opt. Express
**2019**, 27, 3055–3064. [Google Scholar] [CrossRef] - Bouchard, F.; Sit, A.; Hufnagel, F.; Abbas, A.; Zhang, Y.; Heshami, K.; Fickler, R.; Marquardt, C.; Leuchs, G.; Boyd, R.; et al. Underwater Quantum Key Distribution in Outdoor Conditions with Twisted Photons. arXiv
**2018**, 1801, 10299. [Google Scholar] - Ruan, X.; Zhang, H.; Zhao, W.; Wang, X.; Li, X.; Guo, Y. Discrete-Modulated Continuous-Variable Quantum Key Distribution over Seawater Channel. Appl. Sci.
**2019**, 9, 4956. [Google Scholar] [CrossRef] [Green Version] - Kitagawa, A.; Takeoka, M.; Sasaki, M.; Chefles, A. Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states. Phys. Rev. A
**2006**, 73, 042310. [Google Scholar] [CrossRef] [Green Version] - Guo, Y.; Liao, Q.; Wang, Y.; Huang, D.; Huang, P.; Zeng, G. Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction. Phys. Rev. A
**2017**, 95, 032304. [Google Scholar] [CrossRef] [Green Version] - Guo, Y.; Ye, W.; Zhong, H.; Liao, Q. Continuous-variable quantum key distribution with non-Gaussian quantum catalysis. Phys. Rev. A
**2019**, 99, 032327. [Google Scholar] [CrossRef] [Green Version] - Peng, Q.; Chen, G.; Li, X.; Liao, Q.; Guo, Y. Performance Improvement of Underwater Continuous-Variable Quantum Key Distribution via Photon Subtraction. Entropy
**2019**, 21, 1011. [Google Scholar] [CrossRef] [Green Version] - Bennett, C.; Brassard, G.; Mermin, N. Quantum cryptography without Bell’s theorem. Phys. Rev. Lett.
**1992**, 68, 557–559. [Google Scholar] [CrossRef] - Yang, R.H.; He, G.Q. The Influence of Faraday Mirror’s Imperfection in Continuous Variable Quantum Key Distribution System. Acta Photonica Sinica
**2015**, 44, 2. [Google Scholar] - Pirandola, S.; Laurenza, R.; Ottaviani, C.; Banchi, L. Fundamental limits of repeaterless quantum communications. Nat. Commun.
**2017**, 8, 15043. [Google Scholar] [CrossRef] [Green Version] - Tian, B.; Zhang, F.; Tan, X. Design and development of an LED-based optical communication system for autonomous underwater robots. In Proceedings of the 2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Wollongong, Australia, 9–12 July 2013; pp. 1558–1563. [Google Scholar]
- Doniec, M.; Detweiler, C.; Vasilescu, I.; Rus, D. Using optical communication for remote underwater robot operation. In Proceedings of the 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan, 18–22 October 2010; pp. 4017–4022. [Google Scholar]
- Sun, X.; Kang, C.; Kong, M.; Alkhazragi, O.; Guo, Y.; Ouhssain, M.; Weng, Y.; Jones, B.; Ng, T.; Ooi, B. A Review on Practical Considerations and Solutions in Underwater Wireless Optical Communication. OSA
**2020**, 38, 421–431. [Google Scholar] [CrossRef] [Green Version] - Wiscombe, W. Improved Mie scattering algorithms. Appl. Opt.
**1980**, 19, 1505–1509. [Google Scholar] [CrossRef] - Lock, J.; Gérard, G. Generalized Lorenz–Mie theory and applications. J QUANT SPECTROSC RA
**2009**, 110, 800–807. [Google Scholar] [CrossRef] - Zeng, Z.; Fu, S.; Zhang, H.; Dong, Y.; Cheng, J. A Survey of Underwater Optical Wireless Communications. IEEE Commun. Surv. Tutor.
**2017**, 19, 204–238. [Google Scholar] [CrossRef] - Danielewicz-Ferchmin, I. Phase Diagram of Hydration Shells in Ionic Solutions. J. Phys. Chem.
**1995**, 99, 5658–5665. [Google Scholar] [CrossRef] - Shvab, I. Sadus, R. Structure and polarization properties of water: Molecular dynamics with a nonadditive intermolecular potential. Phys. Rev. E
**2012**, 85, 051509. [Google Scholar] [CrossRef] [PubMed] - Coker, H. Empirical free-ion polarizabilities of the alkali metal, alkaline earth metal, and halide ions. J. Phys. Chem.
**1976**, 80, 2078–2084. [Google Scholar] [CrossRef] - Farinato, R.; Rowell, R. New values of the light scattering depolarization and anisotropy of water. J. Chem. Phys.
**1976**, 65, 593. [Google Scholar] [CrossRef] - Duntley, S. Light in the Sea*. J. Opt. Soc. Am.
**1963**, 53, 214–233. [Google Scholar] [CrossRef] - Pegau, W.; Gray, D.; Zaneveld, J. Absorption and attenuation of visible and near-infrared light in water: Dependence on temperature and salinity. OSA
**1997**, 36, 6035–6046. [Google Scholar] [CrossRef] [Green Version] - Haltrin, V. Apparent optical properties of the sea illuminated by sun and sky: Case of the optically deep sea. Appl. Opt.
**1998**, 37, 8336–8340. [Google Scholar] [CrossRef] - Zaneveld, J.; Barnard, Z.; Boss, E. Theoretical derivation of the depth average of remotely sensed optical parameters. Opt. Express
**2005**, 13, 9052–9061. [Google Scholar] [CrossRef] [Green Version] - Guo, Y.; Xie, C.; Huang, P.; Li, J.; Zhang, L.; Huang, D.; Zeng, G. Channel-parameter estimation for satellite-to-submarine continuous-variable quantum key distribution. Phys. Rev. A
**2018**, 97, 052326. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of the zero-photon catalysis (ZPC) based measurement-device-independent-continuous-variable quantum key distribution (MDI-CVQKD) through underwater channel. Hom: homodyne detection, Het: heterodyne detection, PD: photon detector, BS: beam splitter.

**Figure 3.**(

**a**) The secret key rate as a function of ${V}_{A}$ (${V}_{B}$) for the traditional scheme (blue surface) and the ZPC-based (magenta surface) and the single photon subtraction (SPS)-based scheme (green surface). (

**b**) A cross section of (a) where depth is set to 30 m for the traditional (yellow), the ZPC-based (blue), and the SPS-based (red).

**Figure 4.**The secret key rate of the MDI-CVQKD system under pure seawater via the ZPC-based scheme, the SPS-based scheme, and the traditional scheme. T(SPS) = 0.9. The purple line represents PLOB [28] bound. (

**a**). ${V}_{A}={V}_{B}=40$. (

**b**). ${V}_{A}={V}_{B}=150.$

**Figure 5.**The secret key rate of the MDI-CVQKD system under pure seawater for ${V}_{A}={V}_{B}=40$. (

**a**) the ZPC-based scheme, (

**b**) the SPS-based scheme.

**Figure 6.**Relationship among secret key rate, transmission depth and temperature for ${V}_{A}={V}_{B}=40$.

**Figure 7.**Secret key rate of the ZPC-based MDI-CVQKD in oligotrophic seawater under different sun elevation angle for ${V}_{A}={V}_{B}=40$. The upper three lines represent PLOB bound corresponding different sun elevation angle.

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**MDPI and ACS Style**

Wang, Y.; Zou, S.; Mao, Y.; Guo, Y.
Improving Underwater Continuous-Variable Measurement-Device-Independent Quantum Key Distribution via Zero-Photon Catalysis. *Entropy* **2020**, *22*, 571.
https://doi.org/10.3390/e22050571

**AMA Style**

Wang Y, Zou S, Mao Y, Guo Y.
Improving Underwater Continuous-Variable Measurement-Device-Independent Quantum Key Distribution via Zero-Photon Catalysis. *Entropy*. 2020; 22(5):571.
https://doi.org/10.3390/e22050571

**Chicago/Turabian Style**

Wang, Yuang, Shanhua Zou, Yun Mao, and Ying Guo.
2020. "Improving Underwater Continuous-Variable Measurement-Device-Independent Quantum Key Distribution via Zero-Photon Catalysis" *Entropy* 22, no. 5: 571.
https://doi.org/10.3390/e22050571