Long-distance continuous-variable quantum key distribution using separable Gaussian states

Continuous-variable quantum key distribution (CVQKD) is considered to be an alternative to classical cryptography for secure communication. However, its transmission distance is restricted to metropolitan areas, given that it is affected by the channel excess noise and losses. In this paper, we present a scheme for implementing long-distance CVQKD using separable Gaussian states. This tunable QKD protocol requires separable Gaussian states, which are squeezed and displaced, along with the assistance of classical communication and available linear optics components. This protocol originates from the entanglement of one mode and the auxiliary mode used for distribution, which is first destroyed by local correlated noises and restored subsequently by the interference of the auxiliary mode with the second distant separable correlated mode. The displacement matrix is organized by two six-dimensional vectors and is finally fixed by the separability of the tripartite system. The separability between the ancilla and Alice and Bob's system mitigates the enemy's eavesdropping, leading to tolerating higher excess noise and achieving longer transmission distance.

Figure 1

Alice's particle and Bob's particle interact with a mediating particle $C$ continuously. Alice and Bob get entangled while leaving $C$ separable from the system $AB$. WCL denotes weak coherent laser and $S\left(X\right), S\left(P\right)$ are compression operations along position and momentum directions. $D$ is a local displacement distributed according to the Gaussian distribution with correlation matrix $Q$.

Figure 2

Eigenvalues, ${\nu }_{\min }$ and ${\kappa }_{\min }$, as a function displacement parameter $x$, for different compression parameters $\tau$, correspond to the dashed and full lines. The compression parameter $\tau =0.1$ in (a) and $\tau =1$ in (b). The dotted lines denote the boundary of separability.

Figure 3

Scheme of CVQKD by sending separable Gaussian states. Alice and Bob apply displacement operation on their state at the stage of preparation. The displacement ensures the separability between $C$ and $AB$. These modes emerge randomly in phase space and obey Gaussian distribution as shown in the left part. The right part gives the detection scheme. WCL denotes weak coherent laser and $S\left(X\right), S\left(P\right)$ are compression operations along momentum and position directions. Double-headed arrow is local displacement distributed according to the correlation matrix.

Figure 4

Equivalent excess noise as a function of channel transmission $\eta$. The dashed lines are the equivalent excess noise of the original protocol, while the full lines denote the proposed one. From bottom to top, $N_0=1,3,5$.

Figure 5

Secret key rates versus transmission distance from Alice to Bob of the direct reconciliation case. The secret key rate decreases as the transmission distance grows. Simulation results refer to $V=2$ (blue dashed line), $V=10$ (red full line), $V=30$ (blue dashed line), and $V=100$ (green dot-dashed line).

Figure 6

Secret key rates versus channel transmission, $\eta$. The full lines are under the ideal condition with zero excess noise, while the dot-dashed and dotted lines correspond to $N_0=2$ and 4, respectively. The thick and thin lines are under the condition that modulation variance $V=10$ and 100.

Figure 7

Secret key rates versus transmission distance, $L$. The full lines correspond to the condition with excess noise $N_0=1.01$, while the dashed lines correspond to $N_0=2$. The thin lines represent the proposed protocol with separable Gaussian states, while the thick lines are the traditional protocols. In the simulation, the modulation variance $V=30$.

Figure 8

Secret key rates of CVQKD with separable states versus the upper bound of CVQKD. The thick green line is the upper bound of the traditional CVQKD. The dotted, dashed, and thin full lines are the proposed CVQKD protocols with $N_0=1,2,3$, respectively.