# Improving Parameter Estimation of Entropic Uncertainty Relation in Continuous-Variable Quantum Key Distribution

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## Abstract

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## 1. Introduction

## 2. Composable Security and Description of the Protocol

**State preparation**: Alice holds the squeezed states with squeezed variance ${V}_{S}$ before the protocol begins, where ${V}_{S}\in \left(\right)open="("\; close="]">0,1$. In every run of the protocol, Alice uses Gaussian random numbers ${x}_{M}$ to encode the displacement of quadratures by using modulators (generally containing amplitude and phase modulators), and the total modulation variance is denoted by ${V}_{M}$.**State transmission**: Alice sends the modulated state in the quantum channel, which is treated as a totally untrusted channel and controlled by Eve.**State measurement**: Bob receives the quantum state and randomly measures x or p quadrature by an ideal homodyne detector. Resulting from the fact that the practical measurement phase is always discrete, the ideal measurement outcomes should be discretized by the analogue-to-digital converter (ADC). The final discretized results are denoted by ${x}_{B}$.**Parameter estimation**: Alice and Bob repeat the above steps many times until they have enough raw data (e.g., N). Then, Alice or Bob reveals some of the raw data (with length m) through the classical channel to estimate the key parameters of the channel, especially the data distance ${d}_{0}$ between Alice’s and Bob’s data, the transmittance $\tau $, and the excess noise $\epsilon $. See Section 3 for a detailed explanation of the parameter estimation step.**Error correction**: According to the estimation parameters $\tau $ and $\epsilon $, the communication parts estimate the leakage information ${\ell}_{EC}$ during the error correction phase and choose an appropriate classical error reconciliation algorithm, e.g., low-density-parity-check (LDPC) code, to correct Alice’s error (in reverse reconciliation cases) or Bob’s error (in direct reconciliation cases).**Privacy amplification**: Alice and Bob randomly choose a universal${}_{2}$ hash function [45] and apply it to their respective keys to get the final private keys ${s}_{A}$ and ${s}_{B}$ with length ℓ, which are only known to themselves.

## 3. Channel Parameter Estimation with Finite-Size

#### 3.1. Estimation of Smooth Min-Entropy

#### 3.2. Ideal Estimation of Leakage Information with Infinite-Size

#### 3.3. Practical Estimation of Leakage Information with Finite-Size

## 4. Double-Data Modulation Method and the Modified Estimation Process

## 5. Numerical Simulation and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CV | Continuous-variable |

DV | Discrete-variable |

QKD | Quantum key distribution |

EUR | Entropic uncertainty relation |

CM | Covariance matrix |

DR | Direct reconciliation |

RR | Reverse reconciliation |

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**Figure 1.**Prepare-and-measure (PM) scheme of continuous-variable (CV)-quantum key distribution (QKD) using squeezed states. Source: squeezed-state source with squeezed variance ${V}_{S}$; Mod: modulators containing amplitude and phase quadrature modulators with total modulation variance ${V}_{M}$; Hom: homodyne detection; ${x}_{M}$: Gaussian modulation data on Alice’s side; ${x}_{B}$: measurement results on Bob’s side; Quantum channel: channel for the transmission of quantum states, with the transmittance $\tau $ and the excess noise $\epsilon $; Classical channel: channel for the transmission of classical data during the post-processing procedure.

**Figure 2.**Comparison of performances between the previous key rates and the modified results under different block lengths, namely, ${10}^{7}$, ${10}^{8}$, and ${10}^{9}$. (

**a**) shows the direct reconciliation (DR) cases, and (

**b**) shows the reverse reconciliation (RR) cases. The solid lines are the performances under the ideal covariance matrix (CM) estimation, and the dashed lines are the performances under practical CM estimation considering finite-size. The reconciliation efficiency $\beta $ is under a practical value of $95\%$, and the excess noise is chosen as $\epsilon =0.01$. We set the security parameters ${\epsilon}_{c}={\epsilon}_{s}={10}^{-9}$ and the detection range to $\alpha =61.6$.

**Figure 3.**Comparison of performances between the previous key rates and the modified results under different transmission distances. (

**a**) shows the direct reconciliation cases, and (

**b**) shows the reverse reconciliation cases. The solid lines are the performances under ideal CM estimation, and the dashed lines are the performances under practical CM estimation considering finite-size. The parameters are chosen as in Figure 2.

**Figure 4.**Comparison of the performances between the standard estimation method and the modified double-data modulation method under the reverse reconciliation case. (

**a**) shows the performances of two scenarios under different block sizes, while (

**b**) shows the protocol’s performances under different transmission distances. The dashed lines are the performances using the standard estimation method, and the solid lines are the performances using double-data modulation method.

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## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Zhang, Y.; Wang, X.; Yu, S.; Guo, H.
Improving Parameter Estimation of Entropic Uncertainty Relation in Continuous-Variable Quantum Key Distribution. *Entropy* **2019**, *21*, 652.
https://doi.org/10.3390/e21070652

**AMA Style**

Chen Z, Zhang Y, Wang X, Yu S, Guo H.
Improving Parameter Estimation of Entropic Uncertainty Relation in Continuous-Variable Quantum Key Distribution. *Entropy*. 2019; 21(7):652.
https://doi.org/10.3390/e21070652

**Chicago/Turabian Style**

Chen, Ziyang, Yichen Zhang, Xiangyu Wang, Song Yu, and Hong Guo.
2019. "Improving Parameter Estimation of Entropic Uncertainty Relation in Continuous-Variable Quantum Key Distribution" *Entropy* 21, no. 7: 652.
https://doi.org/10.3390/e21070652