Quantum Secret Sharing Based on Continuous-Variable GHZ States

Abstract

Based on the continuous-variable GHZ states, an efficient (n, n) quantum secret sharing protocol is designed, where the Dealer can distribute the various shares to different participants at one time. Likewise, every participant can transfer the same message (share) to other participants simultaneously in the secret recovery process. In this way, the cost of time and quantum photons sharply decreases. What’s more, the proposed QSS is proved to be feasible in the terrible quantum channel with low transmission efficiency and be secure without any secret information leakage under the participant attack.

Appendix: The four-particle CV GHZ states

The CV four-particle GHZ entangled states (a_A, a_B, a_C, a_D)are

$$ \begin{array}{@{}rcl@{}} x_{A} &=& \sqrt{\frac{1}{4}}e^{r}\hat{x}_{4}^{(0)}-\sqrt{\frac{1}{12}}e^{-r}\hat{x}_{3}^{(0)}-\sqrt{\frac{1}{6}}e^{-r}\hat{x}_{2}^{(0)}-\sqrt{\frac{1}{2}}e^{-r}\hat{x}_{1}^{(0)}, \\ p_{A} &=& \sqrt{\frac{1}{4}}e^{-r}\hat{p}_{4}^{(0)}-\sqrt{\frac{1}{12}}e^{r}\hat{p}_{3}^{(0)}-\sqrt{\frac{1}{6}}e^{r}\hat{p}_{2}^{(0)}-\sqrt{\frac{1}{2}}e^{r}\hat{p}_{1}^{(0)}, \\ x_{B} &=& \sqrt{\frac{1}{4}}e^{r}\hat{x}_{4}^{(0)}-\sqrt{\frac{1}{12}}e^{-r}\hat{x}_{3}^{(0)}-\sqrt{\frac{1}{6}}e^{-r}\hat{x}_{2}^{(0)}+\sqrt{\frac{1}{2}}e^{-r}\hat{x}_{1}^{(0)}, \\ p_{B} &=& \sqrt{\frac{1}{4}}e^{-r}\hat{p}_{4}^{(0)}-\sqrt{\frac{1}{12}}e^{r}\hat{p}_{3}^{(0)}-\sqrt{\frac{1}{6}}e^{r}\hat{p}_{2}^{(0)}+\sqrt{\frac{1}{2}}e^{r}\hat{p}_{1}^{(0)}, \\ x_{C} &=& \sqrt{\frac{1}{4}}e^{r}\hat{x}_{4}^{(0)}-\sqrt{\frac{1}{12}}e^{-r}\hat{x}_{3}^{(0)}+\sqrt{\frac{2}{3}}e^{-r}\hat{x}_{2}^{(0)}, \\ p_{C} &=& \sqrt{\frac{1}{4}}e^{-r}\hat{p}_{4}^{(0)}-\sqrt{\frac{1}{12}}e^{r}\hat{p}_{3}^{(0)}+\sqrt{\frac{2}{3}}e^{r}\hat{p}_{2}^{(0)}, \\ x_{D} &=& \sqrt{\frac{1}{4}}e^{r}\hat{x}_{4}^{(0)}+\sqrt{\frac{3}{4}}e^{-r}\hat{x}_{3}^{(0)},\\ p_{D} &=& \sqrt{\frac{1}{4}}e^{-r}\hat{p}_{4}^{(0)}+\sqrt{\frac{3}{4}}e^{r}\hat{p}_{3}^{(0)}, \\ \end{array} $$

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where $a^{(0)}_{j}=\hat {x}_{j}^{(0)}+i\hat {p}_{j}^{(0)}, j=\{1,2,3,4\}$ is a vacuum state and $\hat {x}_{j}^{(0)}, \hat {p}_{j}^{(0)} \sim N(0,1)$ follows the Gaussian distribution. As the squeezed parameter |r| increases, the correlations among all states a_j = x_j + ip_j, j = {A, B, C, D} become increasingly perfect, i.e.,

$$ \begin{array}{@{}rcl@{}} \underset{r\rightarrow +\infty}{\lim} x_{A}-x_{B}&=0,\\ \underset{r\rightarrow +\infty}{\lim} x_{A}-x_{C}&=0,\\ \underset{r\rightarrow +\infty}{\lim} x_{A}-x_{D}&=0,\\ \underset{r\rightarrow +\infty}{\lim} p_{A} +p_{B}+p_{C}+p_{D}&=0. \end{array} $$

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