Estimation of output-channel noise for continuous-variable quantum key distribution

Estimation of channel parameters is important for extending the range and increasing the key rate of continuous-variable quantum key distribution protocols. We propose an estimator for the channel noise parameter based on the method-of-moments. The method-of-moments finds an estimator from the moments of the output distribution of the protocol. This estimator has the advantage of being able to use all of the states shared between Alice and Bob. Other estimators are limited to a smaller publicly revealed subset of the states. The proposed estimator has a lower variance for the high-loss channel than what has previously been proposed. We show that the method-of-moments estimator increases the key rate by up to an order of magnitude at the maximum transmission of the protocol.

Figure 1

Plot of the standard deviation of the different noise estimators vs distance in a fiber channel with a loss of 0.2 dB/km: ${\widehat{V}}_{\xi }$ (dot-dashed line), ${\widehat{\sigma }}_{\mathrm{MM}}^2$ (solid line), ${\widehat{\sigma }}_{\mathrm{MLE}}^2$ (double-dot-dashed line), ${\widehat{V}}_{\xi }^{\mathrm{opt}}$ (dotted line), and ${\widehat{\sigma }}_{\mathrm{opt}}^2$ (dashed line). A stochastic simulation of the coherent-state protocol was repeated 5000 times to obtain the data points for ${\widehat{\sigma }}_{\mathrm{MM}}^2$ (circles) and ${\widehat{\sigma }}_{\mathrm{opt}}^2$ (squares). The parameters used were $V_A=3$, $\xi =0.01$, $m=0.5\times {10}^5$, $N={10}^5$, and $V_{\mathrm{M}2}=10$. The orange double-dot-dashed line is the standard deviation of the MLE with $m=N$ and represents the best estimate Alice and Bob can make of the channel noise using the MLE. The MM estimators and the double-modulation estimators approach this standard deviation as the channel losses increases.

Figure 2

Plot of the key rate with finite key effects related to parameter estimation. The values $V_{\mathrm{A}}$ and $m$ have been optimized with $\xi =0.01$ and $\beta =0.95$ to maximize the key rate using ${\widehat{\sigma }}_{\mathrm{MLE}}^2$ (dotted line), ${\widehat{\sigma }}_{\mathrm{MM}}^2$ (dot-dashed line), and ${\widehat{\sigma }}_{\mathrm{opt}}^2$ (solid line) to estimate the excess noise for (from left to right) $N={10}^5$, $N={10}^7$, $N={10}^9$, and $N={10}^{12}$. The asymptotic key rate with $V_{\mathrm{A}}$ optimized is also plotted (black solid line). As expected, the maximum distance increases with the size of $N$. We see that the optimal estimator outperforms the MLE and MM estimator. As with Fig. 1, we find the MM estimator is worse than the MLE at low channel loss but is better for a lossy channel.

Figure 3

The optimized values of $\frac{m}{N}$ and $V_{\mathrm{A}}$ for the key rates in Fig. 2, where $N={10}^9$ using ${\widehat{\sigma }}_{\mathrm{MLE}}^2$ (dotted line), ${\widehat{\sigma }}_{\mathrm{MM}}^2$ (dot-dashed line), and ${\widehat{\sigma }}_{\mathrm{opt}}^2$ (solid line) to estimate the excess noise. We see that beyond 40 km more states are able to be used in the final key when the optimal or the MM estimator is used.