Improving Continuous-Variable Quantum Key Distribution in a Turbulent Atmospheric Channel via Photon Subtraction

Abstract

The transmission performance of continuous-variable quantum key distribution (CVQKD) in turbulent atmospheric channel is lower than CVQKD in fiber channel due to the large loss of quantum entanglement in turbulent atmospheric channel. In this paper, we propose a new approach to prolong the transmission performance of CVQKD in a turbulent atmospheric channel by using photon subtraction operation since the proposed approach can effectively enhance the entangled state between quantum. Simulation analysis shows that photon subtraction operation can effectively improve the security key rate of continuous-variable quantum key distribution, and the optimal performance can be achieved when only one photon is subtracted.

Appendices

Appendix A: Two-Mode Squeezed State

Recently, researchers have proposed CVQKD protocol for preparing entangled states [34,35,36]. The most widely used are the Gauss quantum continuous variable prepared for the single-mode vacuum squeezed state or the two-mode vacuum squeezed state. The correlation matrix in the single-mode vacuum compression state can be expressed as

$$ {\gamma} =\left( \begin{array}{ll} e^{-2r} & 0 \\0 & e^{2r} \end{array}\right), $$

(20)

where the average value of the state are zero, and the r is squeezing factor. When r > 0, it means that the x is squeezed, otherwise p is squeezed. In this paper, we consider the two-mode vacuum squeezed state based on the Einstein-Podolsky-Rosen (EPR) states. Finally we have the covariance matrix of the state given by

$$ {\gamma}_{EPR} =\left( \begin{array}{ll} \cosh2r\mathbb{I} & \sinh2r\mathbb{Z} \\ \sinh2r\mathbb{Z} & \cosh2r\mathbb{I} \end{array}\right), $$

(21)

where $\mathbb {I}$ and $\mathbb {Z}$ correspond to matrix diag(1,1) and matrix diag(1,− 1), respectively. We take $\gamma _{AB_{0}}$ to describe the combination of two rotated squeezed vacuum states(EPR state) on a balanced beam splitter. Consequently, $\gamma _{AB_{0}}$ can be calculated as

$$ {\gamma}_{AB_{0}}=\left( \begin{array}{ll} V\mathbb{I} & \sqrt{V^{2}-1}\mathbb{Z} \\ \sqrt{V^{2}-1}\mathbb{Z} & V\mathbb{I} \end{array}\right), $$

(22)

where $V =\left (V_{A}+V_{S} \right )/2$ is Alice’s modulation variance, V_A is for the antisqueezed state and V_S is for the squeezed state. Therefore, we have the symbols

$$ X=V, Y=V, Z=\sqrt{V^{2}-1}. $$

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In this scheme, Alice applies two squeezers on the balanced beam splitter to produce a quantum two-mode squeezed state. Then she performs measurements on one mode of them and sends the other mode to Bob. Alice and Bob transmit measurement parameters over the classic channel and establish the secret key after using privacy amplification.

Appendix B: Heterodyne and Homodyne Detection

In homodyne detection case, Bob performs a homodyne detection on the received information, the mutual information $I_{AB}^{Hom}$ (I_(A:B) is expressed as $I_{(A:B)}^{Hom}$) between Alice and Bob can be evaluated as

$$ \begin{array}{@{}rcl@{}} I_{(A:B)}^{Hom}&=& \frac{1}{2}log\frac{V_{A}+1}{V_{(A|B)}^{Hom}+1}\\ &=&\frac{1}{2}log\left (\frac{a+1}{a-\frac{c^{2}}{b}+1} \right ). \end{array} $$

(24)

In addition, $S_{(A|B)}^{Hom}$ (S_(A|B) is expressed as $S_{(A|B)}^{Hom}$) can be calculated as $S_{(A|B)}^{Hom}=G\left (\frac {\lambda _{3}-1}{2} \right )$, where the simplistic eigenvalues ${\lambda _{3}^{2}}(Hom) = a(a-\frac {c^{2}}{b})$. When Bob uses the homodyne detection, the resulting secret key rate is finally expressed as

$$ K^{Hom}=P_{\left (k \right )}^{\widehat{\varPi}_{1}}\left \{\beta I_{(A:B)}^{Hom}-[S_{(AB)}-S_{(A|B)}^{Hom}] \right \}. $$

(25)

In heterodyne detection case, when Bob performs a heterodyne detection on the received information, the mutual information $I_{AB}^{Het}$ (I_(A:B) is expressed as $I_{(A:B)}^{Het}$) between Alice and Bob can be evaluated as

$$ \begin{array}{@{}rcl@{}} I_{A:B}^{Het}&=&log\frac{V_{B}+1}{V_{B|A}^{Het}+1}\\ &=&log\left (\frac{b+1}{\left( b-\frac{c^{2}}{a+1}\right)+1} \right ). \end{array} $$

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Subsequently, $S_{(A|B)}^{Het}$ (S_(A|B) is expressed as $S_{(A|B)}^{Het}$) can be calculated as

$$ S_{(A|B)}^{Het}=G\left (\frac{\lambda_{3}-1}{2} \right ), $$

(27)

where the simplistic eigenvalues are given by

$$ \lambda_{3}(Het)=a-\frac{c^{2}}{b+1}. $$

(28)

When Bob applies the heterodyne detection, the resulting secret key rate is expressed as

$$ K^{Het}=P_{\left (k \right )}^{\widehat{\varPi }_{1}}\left \{\beta I_{(A:B)}^{Het}-\left[S_{(AB)}-S_{(A|B)}^{Het}\right] \right \}. $$

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