# Security Analysis of Unidimensional Continuous-Variable Quantum Key Distribution Using Uncertainty Relations

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## Abstract

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## 1. Introduction

## 2. Unidimensional Quantum Key Distribution

#### 2.1. Equivalence between the EB Scheme and the PM Scheme

_{M}, which is assumed to be expressed in shot-noise units, and that the coherent states follow the uncertainty principle of variance 1. Thus, the mixture of Gaussian-modulated coherent states gives rise to a unidimensional chain structure with a thickness of 1 and a length of $\sqrt{1+{V}_{M}}$ in the phase space. These quantum states are then sent to Bob through an untrusted quantum channel with transmittance ${T}_{x}$, ${T}_{y}$ and excess noise ${\epsilon}_{x}$, ${\epsilon}_{y}$.

#### 2.2. Calculation of Secret Key Rate with Reverse Reconciliation

## 3. Security Analysis Using Uncertainty Relations

#### 3.1. Uncertainty Relations for Symmetrical Coherent-State Protocol

#### 3.2. Uncertainty Relations for Unidimensional Coherent-State Protocol

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Unidimensional (UD) protocol schemes under realistic conditions. (

**a**) Prepare-and-measure (PM) scheme of the UD protocol; (

**b**) Entanglement-based (EB) scheme of the UD protocol.

**Figure 3.**(

**a**) Secure and unsecure regions of the SY protocol using ideal homodyne detector ($\eta =1,{v}_{el}=0$) and realistic homodyne detector ($\eta =0.6,{v}_{el}=0.1$); (

**b**) Secret key rate versus the excess noise for different channel losses.

**Figure 4.**Comparison among physical regions of the UD protocol under both ideal and realistic detection conditions. The red solid line represents the realistic parabolic curve (equivalent to Bob using a realistic homodyne detector with $\eta =0.6,{v}_{el}=0.1$) and black solid line is the ideal parabolic curve (equivalent to Bob using an ideal homodyne detector with $\eta =1,{v}_{el}=0$). The red dashed line represents the part where the key rate is zero under realistic detection condition. Here, we set: $\beta =0.99$, ${T}_{x}=0.4$ (corresponding to a distance of 20 km fiber), ${\epsilon}_{x}=0.01$ and ${V}_{M}=6.35$.

**Figure 5.**Comparison among physical regions delimited by the parabolic curves of the UD protocol. The black solid curve corresponds to the ideal parabolic curve, whereas the others to the realistic parabolic curves obtained for different parameter conditions. (

**a**) Changes of the physical region extension according to different values of $\eta $ (${v}_{el}$ remains constant); (

**b**) Changes of the physical region extension according to different values of ${v}_{el}$ ($\eta $ remains constant). The values of the parameters ${T}_{x}$, ${\epsilon}_{x}$, and ${V}_{M}$ are the same as in Figure 4.

**Figure 6.**Ideal parabolic curve versus related parameters. (

**a**) Different modulation variance values with ${T}_{x}=0.1$ and ${\epsilon}_{x}=0.01$; (

**b**) Different transmission efficiency values with ${\epsilon}_{x}=0.01$ and ${V}_{M}=3$; (

**c**) Different excess noise values with ${T}_{x}=0.1$ and ${V}_{M}=3$; (

**d**) Different reconciliation efficiency values with ${T}_{x}=0.1$, ${\epsilon}_{x}=0.01$, and ${V}_{M}=3$.

**Figure 7.**Comparison among physical regions delimited by the new curves of the UD protocol. (

**a**) Changes of the physical region according to different values of $\eta $ (${v}_{el}$ remains constant); (

**b**) Changes of the physical region according to different values of ${v}_{el}$ ($\eta $ remains constant). The other parameters are $\beta =0.99$, ${T}_{x}=0.1$, ${\epsilon}_{x}=0.01$, and ${V}_{M}=3$.

**Figure 8.**Secure and unsecure regions of the UD protocol under realistic detection condition. The parameters are set to $\beta =0.99$, ${V}_{M}=3$, ${T}_{x}=0.1$, ${\epsilon}_{x}=0.01$, $\eta =0.6$, and ${v}_{el}=0.1$.

**Figure 9.**(

**a**) Comparison between secure and unsecure regions for the SY coherent-state protocol and UD coherent-state protocol under different detection conditions; (

**b**) Secure and unsecure regions of the UD protocol using an ideal homodyne detector ($\eta =1,{v}_{el}=0$) and a realistic one ($\eta =0.6,{v}_{el}=0.1$). Here we consider ${V}_{M}=3$, $\beta =0.99$, and the estimated value ${V}_{y}^{{B}_{1}}\approx 1+{T}_{x}{\epsilon}_{x}$.

**Figure 10.**(

**a**) Minimum secret key rate as a function of the channel losses; (

**b**) Optimal choice of ${\chi}_{\mathrm{hom}}$ that maximizes the secret key rate in (

**a**). The other parameters are $\beta =0.99$, ${\epsilon}_{x}=0.04$, ${V}_{M}=3$, and ${V}_{y}^{{B}_{1}}\approx 1+{T}_{x}{\epsilon}_{x}$.

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**MDPI and ACS Style**

Wang, P.; Wang, X.; Li, Y.
Security Analysis of Unidimensional Continuous-Variable Quantum Key Distribution Using Uncertainty Relations. *Entropy* **2018**, *20*, 157.
https://doi.org/10.3390/e20030157

**AMA Style**

Wang P, Wang X, Li Y.
Security Analysis of Unidimensional Continuous-Variable Quantum Key Distribution Using Uncertainty Relations. *Entropy*. 2018; 20(3):157.
https://doi.org/10.3390/e20030157

**Chicago/Turabian Style**

Wang, Pu, Xuyang Wang, and Yongmin Li.
2018. "Security Analysis of Unidimensional Continuous-Variable Quantum Key Distribution Using Uncertainty Relations" *Entropy* 20, no. 3: 157.
https://doi.org/10.3390/e20030157