# Performance Analysis of Continuous Variable Quantum Teleportation with Noiseless Linear Amplifier in Seawater Channel

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## Abstract

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## 1. Introduction

## 2. The NLA-Based CVQT under Seawater

#### 2.1. The NLA-Based CVQT Protocol

#### 2.2. Optical Propagation Characteristics of Seawater Channel

## 3. Effect of Excess Noise

## 4. Performance Analysis

#### 4.1. The Gain of NLA under Seawater Channel

#### 4.2. Performance Improvement under Seawater Lossy Channel

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Absorption Coefficient of the Seawater Channel

Parameters | Value | Description |
---|---|---|

${a}_{f}$ | $35.959\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{mg}$ | Fulvic acid spectral absorption coefficient |

${k}_{f}$ | $0.0189\phantom{\rule{3.33333pt}{0ex}}{\mathrm{nm}}^{-1}$ | Fulvic acid absorption slope coefficient |

${a}_{h}$ | $18.828\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{mg}$ | Humic acid spectral absorption coefficient |

${k}_{h}$ | $0.01105\phantom{\rule{3.33333pt}{0ex}}{\mathrm{nm}}^{-1}$ | Humic acid absorption slope coefficient |

${a}_{d}$ | $9.721\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{mg}$ | Mineral and detritus spectral absorption coefficient |

${k}_{d}$ | $0.012\phantom{\rule{3.33333pt}{0ex}}{\mathrm{nm}}^{-1}$ | Mineral and detritus absorption slope coefficient |

## Appendix B. The Scattering Coefficient of the Seawater Channel

Function | Formula |
---|---|

${b}_{w}\left(\lambda \right)$ | $0.005826{(400/\lambda )}^{4.3222}$ |

${b}_{s}\left(\lambda \right)$ | $1.1513{(400/\lambda )}^{1.7}$ |

${C}_{s}\left(d\right)$ | $0.01739{C}_{c}\left(d\right)exp\left(0.11631{C}_{c}\left(d\right)\right)$ |

${b}_{l}\left(\lambda \right)$ | $0.3411{(400/\lambda )}^{0.3}$ |

${C}_{l}\left(d\right)$ | $0.76284{C}_{c}\left(d\right)exp\left(0.03092{C}_{c}\left(d\right)\right)$ |

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**Figure 1.**(Color online) EB version of the NLA-CVQT, where the EPR state is prepared by Alice and ${B}_{0}$ is transmitted to Bob through the seawater lossy channel. At the receiver, Bob use noiseless linear amplifiers for performance improvement. LO, local oscillator; BS, beam splitter; ${g}_{N}$, gain of NLA.

**Figure 2.**(Color online) (

**a**) The functional relationship between absorption coefficient $a({\lambda}_{L},d)$, depth d, wavelength ${\lambda}_{L}$. (

**b**) The functional relationship between scattering coefficient $b({\lambda}_{L},d)$, depth d, wavelength ${\lambda}_{L}$. (

**c**) The functional relationship between total channel attenuation $c({\lambda}_{L},d)$, depth d, wavelength ${\lambda}_{L}$. (

**d**) The functional relationship between transmittance ${T}_{sea}$, depth d, transmission distance L. Here, the variation of absorption coefficient $a({\lambda}_{L},d)$, scattering coefficient $b({\lambda}_{L},d)$, total attenuation coefficient $c({\lambda}_{L},d)$ and transmittance ${T}_{sea}$ are characterized by the change of color, the unit of $a({\lambda}_{L},d)$, $b({\lambda}_{L},d)$, $c({\lambda}_{L},d)$ are m${}^{-1}$.

**Figure 4.**Variation of achievable NLA gain coefficient with transmission distance in seawater lossy channel.

**Figure 5.**Optimal inherent parameters ${\lambda}_{opt}$ as a function of transmission distance and depth in the case of a lossy channel. Here, the NLA gain coefficient $g=2$.

**Figure 6.**(Color online) Fidelity as a function of transmission distance and depth in the case of a lossy channel (

**a**) EPR state parameter $\lambda $ = 0.5, NLA gain coefficient $g\in \{2.7,1.8\}$. (

**b**) EPR state parameter $\lambda $ = 0.7, NLA gain coefficient $g\in \{2.0,1.6\}$.

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**MDPI and ACS Style**

Wu, H.; Liu, X.; Zhang, H.; Ruan, X.; Guo, Y.
Performance Analysis of Continuous Variable Quantum Teleportation with Noiseless Linear Amplifier in Seawater Channel. *Symmetry* **2022**, *14*, 997.
https://doi.org/10.3390/sym14050997

**AMA Style**

Wu H, Liu X, Zhang H, Ruan X, Guo Y.
Performance Analysis of Continuous Variable Quantum Teleportation with Noiseless Linear Amplifier in Seawater Channel. *Symmetry*. 2022; 14(5):997.
https://doi.org/10.3390/sym14050997

**Chicago/Turabian Style**

Wu, Hao, Xu Liu, Hang Zhang, Xinchao Ruan, and Ying Guo.
2022. "Performance Analysis of Continuous Variable Quantum Teleportation with Noiseless Linear Amplifier in Seawater Channel" *Symmetry* 14, no. 5: 997.
https://doi.org/10.3390/sym14050997