# Phase-Matching Continuous-Variable Measurement-Device-Independent Quantum Key Distribution

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## Abstract

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## 1. Introduction

## 2. DMPM CV-MDI-QKD Protocol

## 3. Eavesdropping and Simulations

## 4. Proof-of-Principle Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Secret Key Rate under BS-Combined SD Attacks

**Figure A1.**(Color online). (

**a**) The description of the SD attack strategy on the phase-matching CV-MDI-QKD scheme. BS is beam splitter, PIA is phase-insensitive amplifier, ${\mathrm{Hom}}_{0}$ is homodyne detection of measuring the X quadrature, ${\mathrm{Hom}}_{1}$ is homodyne detection of measuring the P quadrature, T is the transmission efficiency of quantum channel between Alice (Bob) and Charlie; (

**b**) the construction of SD attack, NLA is noiseless linear amplifier.

#### Appendix A.1. The Mutual Information ${I}_{BE}^{BS}$ between Alice and Bob

#### Appendix A.2. The Mutual Information ${I}_{BE}^{BS}$ between Eve and Bob

#### Appendix A.3. Simulations of Secret Key Rate under BS-Combined SD Attacks

**Figure A2.**(Color online). The secret key rates for different channel excess noises against the BS-combined SD attacks for the symmetric case when the SD receivers reach SQL (thick curves) and QL (thin curves), respectively. Solid, dashed, dotted, and dash-dotted curves represent the channel excess noise ${\epsilon}_{c}$ = 0.02, 0.05, 0.08, and 0.1, respectively. The other parameters are set as $\alpha =3.3,\eta =0.6,{\nu}_{el}=0.04,\beta =0.98,g=1,\kappa =0.01\sqrt{2\eta T}$.

**Figure A3.**(Color online). The secret key rates as a function of amplitude $\alpha $ against the BS-combined SD attacks with transmission distance of 3 km of the standard single mode fiber for the symmetric case, when the SD receivers reach QL and SQL, respectively. The other parameters are set as ${\epsilon}_{c}=0.03,$ $\eta =0.6,{\nu}_{el}=0.04,\beta =0.98,g=1,\kappa =0.01\sqrt{2\eta T}$.

**Figure A4.**(Color online). The secret key rates for different $\kappa $ against the BS-combined SD attacks for the symmetric case when the SD receivers reach QL and SQL, respectively. The solid, dashed, dotted, and dash-dotted curves represent $\kappa =0.1\sqrt{2\eta T}$, $0.01\sqrt{2\eta T}$, $0.005\sqrt{2\eta T}$ and $0.001\sqrt{2\eta T}$, respectively. The other parameters are set as ${\epsilon}_{c}=0.03,\eta =0.6,{\nu}_{el}=0.04,\beta =0.98,g=1,\alpha =3.3$.

**Figure A5.**(Color online). The secret key rates for different channel excess noises against the BS-combined SD attacks for the ideally asymmetric case when the SD receivers reach QL (thick curves) and SQL (thin curves), respectively. Solid, dashed, dotted, and dash-dotted curves represent the channel excess noise ${\epsilon}_{c}$ = 0.02, 0.05, 0.08, and 0.1, respectively. The other parameters are set as $\alpha =0.8,\eta =0.6,{\nu}_{el}=0.04,\beta =0.98,g=1,\kappa =\sqrt{2\eta T}$.

**Figure A6.**(Color online). The secret key rates as a function of amplitude $\alpha $ against the BS-combined SD attacks with transmission distance of 50 km of standard single mode fiber for the ideally asymmetric case. The black dashed and red solid curves denote the secret key rates when the SD receivers reach QL and SQL, respectively. The other parameters are set as ${\epsilon}_{c}=0.03,\eta =0.6,{\nu}_{el}=0.04,$ $\beta =0.98,g=1,\kappa =\sqrt{2\eta T}$.

**Figure A7.**(Color online). The secret key rates for different $\kappa $ against the BS-combined SD attacks for the ideally asymmetric case when the SD receivers reach QL and SQL, respectively. The solid, dashed, dotted, and dash-dotted curves represent $\kappa =2\sqrt{2\eta T}$, $\sqrt{2\eta T}$, $0.1\sqrt{2\eta T}$ and $0.01\sqrt{2\eta T}$, respectively. The other parameters are set as ${\epsilon}_{c}=0.03,\eta =0.6,{\nu}_{el}=0.04,\beta =0.98,g=1,\alpha =0.8$.

## Appendix B. Eve’s BS-Combined SD Attack Strategy

**Figure A8.**(Color online). Equivalent realization of beam splitting and amplification operation for Eve’s optimized SD attack.

## Appendix C. Secret Key Rate under Complete IR Attacks

**Figure A9.**(Color online). The description of the IR attack on the symmetric case of DMPM CV-MDI-QKD scheme. BS is the 50:50 beam splitter, $\mathrm{Hom}$ is homodyne detection of measuring the X or P quadrature with quantum efficiency ${\eta}_{e}=1$. These components are all controlled by Eve without reproduction of quantum states. Eve will guess Bob’s encoding state to capture the secret key.

#### Appendix C.1. The Mutual Information between Alice and Bob

#### Appendix C.2. The Mutual Information between Eve and Bob

#### Appendix C.3. The Evaluation of Extra Excess Noise

#### Appendix C.4. Simulations of Secret Key Rate under Complete IR Attacks

**Figure A10.**(Color online). The secret key rates under the complete IR attacks. The other parameters are set as ${\epsilon}_{c}=0.03,\eta =0.6,{\nu}_{el}=0.04,\beta =0.98,\kappa =0.01\xb7\left(2\eta T\right)$.

## Appendix D. Secret Key Rate of the Phase-Matching Protocol

## Appendix E. Frequency Offset Recovery and Phase Drift Compensation

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**Figure 1.**(Color online). The phase-matching CV-MDI-QKD scheme. RNG is random number generator, QM is quadrature phase-shift keying (QPSK) modulator, VOA is variable optical attenuator, BS is beam splitter, Hom${}_{0}$ is homodyne detection of measuring the X quadrature, Hom${}_{1}$ is homodyne detection of measuring the P quadrature, ${T}_{A\left(B\right)}$ and ${\u03f5}_{A\left(B\right)}$ are the transmission efficiency and excess noise of quantum channel between Alice (Bob) and Charlie, respectively.

**Figure 2.**(Color online). The secret key rates for different channel excess noises when the SD receivers reach QL and SQL, respectively. The results are compared to the PLOB bound [9]. The other parameters are set as $\eta =0.9,\alpha =1.1,\kappa =0.25\sqrt{2\eta T}$ (for symmetric case), $\eta =0.6,\alpha =2.5,$$\kappa =0.5\sqrt{2\eta T}$ (for ideally asymmetric case), ${\nu}_{el}=0.04,\beta =0.98,g=1$.

**Figure 3.**(Color online). The RDEs as a function of transmission distance for ideally asymmetric case when the SD receivers reach QL and SQL. The other parameters are set as $\alpha =2.5,{\epsilon}_{c}=0.02,$${\nu}_{el}=0.04,\beta =0.98,g=1,\kappa =0.5\sqrt{2\eta T}$.

**Figure 4.**(Color online). The schematic diagram of the proof-of-principle experiment of the proposed phase-matching CV-MDI-QKD scheme. ${L}_{1}$, ${L}_{2}$: the lasers with stabilized frequencies, BS: beamsplitter, VOA: variable optical attenuator, SMF: single mode fiber, PC: polarization controller, PM: phase modulator, Hom: homodyne detector, PD: photoelectric detector, OSC: oscilloscope.

**Figure 5.**(Color online). (

**a**) Phase drift angle and phase drift rate change with time. (

**b**) The excess noise and corresponding secret key rates for the ideally asymmetric case as a function of the size of frame. The other parameters are set as $\eta =0.6,{\nu}_{el}=0.04,\beta =0.98,g=1,\kappa =0.5\sqrt{2\eta T}$.

Encoded States | Encoded Bits | Decoded Bits |
---|---|---|

($|\alpha \rangle $, $|\alpha \rangle $) | $(0,0)$ | $(0,0)$ |

($|\alpha {e}^{\pi i}\rangle $, $|\alpha {e}^{\pi i}\rangle $) | $(1,1)$ | $(1,1)$ |

($|\alpha {e}^{\pi i/2}\rangle $, $|\alpha {e}^{3\pi i/2}\rangle $) | $(0,1)$ | $(0,0)$ |

($|\alpha {e}^{3\pi i/2}\rangle $, $|\alpha {e}^{\pi i/2}\rangle $) | $(1,0)$ | $(1,1)$ |

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**MDPI and ACS Style**

Huang, P.; Wang, T.; Huang, D.; Zeng, G.
Phase-Matching Continuous-Variable Measurement-Device-Independent Quantum Key Distribution. *Symmetry* **2022**, *14*, 568.
https://doi.org/10.3390/sym14030568

**AMA Style**

Huang P, Wang T, Huang D, Zeng G.
Phase-Matching Continuous-Variable Measurement-Device-Independent Quantum Key Distribution. *Symmetry*. 2022; 14(3):568.
https://doi.org/10.3390/sym14030568

**Chicago/Turabian Style**

Huang, Peng, Tao Wang, Duan Huang, and Guihua Zeng.
2022. "Phase-Matching Continuous-Variable Measurement-Device-Independent Quantum Key Distribution" *Symmetry* 14, no. 3: 568.
https://doi.org/10.3390/sym14030568