Improving the Discrete-Modulated Continuous-Variable Measurement-Device-Independent Quantum Key Distribution with Quantum Scissors

Abstract

We propose a new scheme to strengthen the performance of the discrete-modulated continuous-variable measurement-device-independent quantum key distribution (DM-CV-MDI-QKD) protocol by introducing a quantum scissors (QS) operation to Bob’s side. According to the equivalent one-way scheme of the protocol, we further investigate the success probability of the QS operation, and analyze the security of the DM-CV-MDI-QKD equipped with this operation in the asymptotic case. Simulation results show that the QS operation can indeed effectively improve the performance of the existing protocol, and compared with the scheme without QS operation, our proposed scheme has a higher secret key rate and a longer secure transmission distance.

Appendix: The Expressions of the Parameter in the Covariance Matrix \({\varGamma }_{A_{1} B_{1}^{\prime }}\)

From [26], we can know that the expressions of the parameter in the covariance matrix ${\varGamma }_{A_{1} B_{1}^{\prime }}$ are as follows

$$ \begin{array}{@{}rcl@{}} V_{X}^{\prime} &=&1+\frac{\alpha^{2}}{{\varsigma}} \left\{ \delta_{1} [-A\sinh \left( \frac{T\alpha^{2}}{2F+1}\right)+B\cosh \left( \frac{T\alpha^{2}}{2F+1}\right) \right.\\ &&\left.+C\sinh \left( \frac{T\alpha^{2}}{2F}\right)\right] + \delta_{2} \left[ A\cosh \left( \frac{T\alpha^{2}}{2F+1}\right)-B\sinh \left( \frac{T\alpha^{2}}{2F+1}\right) \right. \\ &&\left.-C\cosh \left( \frac{T\alpha^{2}}{2F}\right)\right] + \delta_{3} \left[ -A\sin \left( \frac{T\alpha^{2}}{2F+1}\right)+B\cos \left( \frac{T\alpha^{2}}{2F+1}\right) \right.\\ &&\left.\left.+C\sin \left( \frac{T\alpha^{2}}{2F}\right) \right]\right/2 - \delta_{4} [ A\cos \left( \frac{T\alpha^{2}}{2F+1}\right)+B\sin \left( \frac{T\alpha^{2}}{2F+1}\right) \\ &&\left.\left.\left.-C\cos \left( \frac{T\alpha^{2}}{2F}\right)\right]\right/2 \right\}, \end{array} $$

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$$ V_{Y}^{\prime}=\frac{1}{{\varsigma}} \left( b_{k} e^{-\frac{T|\alpha_{k}|^{2}}{2F+1}} -\frac{2(1-\mu)}{F}e^{-\frac{T|\alpha_{k}|^{2}}{2F}} \right) -1, $$

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$$ \begin{array}{@{}rcl@{}} V_{XY}^{\prime} &=& \sqrt{T} Z_{4}^{QS} \\ &=&\frac{4\alpha^{2} \sqrt{\mu(1-\mu)T}}{\mathcal{P}_{\text{succ}}(2F+1)^{2}} \left\{\omega_{1} \cosh\left( \frac{T\alpha^{2}}{2F+1}\right)-\omega_{2} \sinh\left( \frac{T\alpha^{2}}{2F+1}\right) \right.\\ &&\left.+\omega_{3} \cos\left( \frac{T\alpha^{2}}{2F+1}\right) -\omega_{4} \sin\left( \frac{T\alpha^{2}}{2F+1} \right) \right\} , \end{array} $$

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where

$$ \begin{array}{@{}rcl@{}} {\varsigma}&=&\frac{2e^{\frac{-T\alpha_{k} \alpha_{l}^{*} }{2F+1}}}{(2F+1)^{3}} \left( (2F+1)^{2}-\mu(2F+1)+\mu T \alpha_{k} \alpha_{l}^{*} \right) \\ &&-\frac{1-\mu}{2F} e^{\frac{-T\alpha_{k} \alpha_{l}^{*} }{2F}} ,\\ A&=&\frac{2}{(2F+1)^{3}} \left( (2F+1)^{2}-\mu(2F+1)\right),\\ B&=&\frac{2\mu T \alpha^{2}}{(2F+1)^{3}}, C=\frac{1-\mu}{2F}, F=\frac{1}{2}+\frac{1}{4} T \epsilon^{\prime} ,\\ \delta_{1}&=&\frac{\lambda_{0}}{\lambda_{1}}+\frac{\lambda_{2}}{\lambda_{3}}, \delta_{2}=\frac{\lambda_{1}}{\lambda_{2}}+\frac{\lambda_{3}}{\lambda_{0}} , \\ \delta_{3}&=&\frac{\lambda_{0}}{\lambda_{1}}-\frac{\lambda_{2}}{\lambda_{3}}, \delta_{4}=\frac{\lambda_{1}}{\lambda_{2}}-\frac{\lambda_{3}}{\lambda_{0}} , \\ \end{array} $$

$$ \begin{array}{@{}rcl@{}} b_{k}&=&\frac{8}{(2F+1)^{3}}\left( (2F+1)^{2}-\mu(2F^{2}+3F+1)+\mu T |\alpha_{k}|^{2} \right), \\ \omega_{1}&=&\sqrt{\frac{\lambda_{0}}{\lambda_{1}}}+\sqrt{\frac{\lambda_{2}}{\lambda_{3}}} , \omega_{2}=\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}+\sqrt{\frac{\lambda_{3}}{\lambda_{0}}} ,\\ \omega_{3}&=&\sqrt{\frac{\lambda_{0}}{\lambda_{1}}}-\sqrt{\frac{\lambda_{2}}{\lambda_{3}}} , \omega_{4}=\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}-\sqrt{\frac{\lambda_{3}}{\lambda_{0}}} . \end{array} $$

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