Secret key rate of a continuous-variable quantum-key-distribution scheme when the detection process is inaccessible to eavesdroppers

We have developed a method to calculate a secret key rate of a continuous-variable quantum-key-distribution scheme using four coherent states and postselection for a general model of Gaussian attacks. We assume that the transmission line and detection process are described by a pair of Gaussian channels. In our analysis, while the loss and noise on the transmission line are induced by an eavesdropper, Eve, who can replace the transmission line with a lossless and noiseless optical fiber, she is assumed inaccessible to the detection process. By separating the transmission noise and detection noise, we can always extract a larger key compared with the case that all loss and noises are induced by an eavesdropper's interference. An asymptotic key rate against collective Gaussian attacks can be determined numerically for the given channels' parameters. The improvement of the key rates turns out to be more significant for the reverse-reconciliation scheme.

Figure 1

(a) Alice sends a coherent state $\left|\alpha \right.〉$ through a lossy and noisy channel. Bob observes quadrature with the total transmission $\eta$ and the total excess noise $\xi$. (b) The transmission channel is modeled by a lossy and noisy Gaussian channel with the transmission ${\eta }_1$ and the excess noise ${\xi }_1$. Bob's detector is modeled by another lossy and noisy Gaussian channel with the transmission ${\eta }_2$ and the excess noise ${\xi }_2$ followed by an ideal homodyne detector. (c) The action of Gaussian channels $({\eta }_i,{\xi }_i)$ with $i=1,2$ can be described by beam-splitter unitaries $U_{B{E}_2}$ and $U_{B{D}_2}$ coupling to two-mode squeezed states ${\left|{\mathrm{\Phi }}_1\right.〉}_E$ and ${\left|{\mathrm{\Phi }}_2\right.〉}_D$.

Figure 2

Key rates for the direct-reconciliation (DR) scheme and the reverse-reconciliation (RR) scheme with the photon number ${\alpha }^2=0.5$ and the total excess noise $\xi =0.01$. The amount of detector's excess noise ${\xi }_2$ is set to 0, 30%, 50%, and 80% of the total excess noise $\xi =0.01$. The circles on the curve of ${\xi }_2=0$ are calculated from the analytic result in Ref. [43].

Figure 3

Key rates for the direct-reconciliation (DR) scheme as functions of distance with the total excess noise $\xi =\{0.005,0.01,0.02\}$. The photon number ${\alpha }^2$ is chosen to maximize the key rate with 0.05 steps. The amount of the detector's excess noise ${\xi }_2$ is set to 0, 30%, 50%, and 80% of the total excess noise $\xi$.

Figure 4

Key rate for the reverse-reconciliation (RR) scheme as functions of distance with the total excess noise $\xi =\{0.005,0.01,0.02\}$. The photon number ${\alpha }^2$ is chosen to maximize the key rate with 0.05 steps. The amount of the detector's excess noise ${\xi }_2$ is set to 0, 30%, 50%, and 80% of the total excess noise $\xi$.