# Unidimensional Two-Way Continuous-Variable Quantum Key Distribution Using Coherent States

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## Abstract

**:**

## 1. Introduction

## 2. The Schemes of Unidimensional Two-Way CV-QKD Protocol

## 3. Security Analysis of the Protocol Against Two-Mode Attack

#### 3.1. Two-Mode Attack Strategy

#### 3.2. The Secret Key Rate of the Protocol

## 4. Simulation and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(Color online) (

**a**) The prepare-and-measure (PM) scheme of the unidimensional (UD) two-way protocol. Here, the two quantum channels are fully controlled by Eve while she has no access to the apparatuses in Alice’s and Bob’s stations. (

**b**) Symmetrical Gaussian modulated coherent state on a phase space. (

**c**) UD modulated coherent state on a phase space, assuming only x quadrature was modulated, which means, the coherent state can only be shifted along x on the phase space. Here, ${x}_{A}$ and ${p}_{A}$ are two independent identically distributed Gaussian random variables, and the coordinates describe the centers of the shifted coherent states. (BS: beam splitter, AM: amplitude modulator, PM: phase modulator).

**Figure 2.**(Color online) The entanglement-based (EB) scheme of the UD two-way protocol against two-mode collective attacks, where Eve has full control of the quantum channels while she has no access to the apparatuses in Alice’s and Bob’s stations. At Alice’s side, one mode of the EPR pair is measured by homodyne detection while the other mode is sent to a squeezer, this part is equivalent to the PM scheme of UD modulation with coherent state. The blue beam splitter with a transmittance of ${T}_{A}$ is used to couple Alice’s state with Bob’s state. (QM: quantum memory, ${B}_{0}$ vacuum state).

**Figure 3.**(Color online) Secret key rate of UD two-way CV-quantum key distribution (QKD) protocol under all accessible two-mode attacks under different situations. (

**a**) At 5 km while $V=5$, $\beta =0.956$, $\u03f5=0.04$. (

**b**) At 5 km while $V={10}^{3}$, $\beta =1$, $\u03f5=0.04$. (

**c**) At 10 km while $V=5$, $\beta =0.956$, $\u03f5=0.04$. (For all simulations ${T}_{A}=0.8$).

**Figure 4.**(Color online) Comparisons of secret key rates between UD two-way CV-QKD protocol (solid line) and a sub-protocol ($Het-Ho{m}_{M}$) of the conventional symmetrical Gaussian modulated two-way CV-QKD protocol family [20] (dashed line), under ideal and practical situations. The attack strategies against both protocols are one-mode attack. (

**a**) Ideal situations where $V={10}^{3}$, $\beta =1$, $\u03f5=0.04$ (red line), $0.045$ (blue line) and $0.05$ (black line). (

**b**) Practical situations where $V=5$, $\beta =0.956$, $\u03f5=0.04$ (red line), $0.045$ (blue line), $0.05$ (black line). (For both protocols ${T}_{A}=0.8$).

**Figure 5.**(Color online) Comparison of tolerable excess noise between UD two-way CV-QKD protocol (solid lines) and the sub-protocol ($Het-Ho{m}_{M}$) of the conventional symmetrical Gaussian modulated two-way protocol family (dashed lines), the red lines represent the ideal situations and the blue lines represent the practical situations. (The attack strategies against both protocols are one-mode attack, and for both protocols ${T}_{A}=0.8$).

**Figure 6.**(Color online) Comparison of secret key rate between UD two-way CV-QKD protocol against the optimal two-mode attack (red line) and the one-mode attack (black line). Here, the solid line represents the practical situation, the dashed line and the dotted line represent the ideal situation. (For both cases ${T}_{A}=0.8$).

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**MDPI and ACS Style**

Bian, Y.; Huang, L.; Zhang, Y.
Unidimensional Two-Way Continuous-Variable Quantum Key Distribution Using Coherent States. *Entropy* **2021**, *23*, 294.
https://doi.org/10.3390/e23030294

**AMA Style**

Bian Y, Huang L, Zhang Y.
Unidimensional Two-Way Continuous-Variable Quantum Key Distribution Using Coherent States. *Entropy*. 2021; 23(3):294.
https://doi.org/10.3390/e23030294

**Chicago/Turabian Style**

Bian, Yiming, Luyu Huang, and Yichen Zhang.
2021. "Unidimensional Two-Way Continuous-Variable Quantum Key Distribution Using Coherent States" *Entropy* 23, no. 3: 294.
https://doi.org/10.3390/e23030294