Reverse-reconciliation continuous-variable quantum key distribution based on the uncertainty principle

A big challenge in continuous-variable quantum key distribution is to prove security against arbitrary coherent attacks including realistic assumptions such as finite-size effects. Recently, such a proof has been presented in [Phys. Rev. Lett. 109, 100502 (2012)] for a two-mode squeezed state protocol based on a novel uncertainty relation with quantum memories. But the transmission distances were fairly limited due to a direct reconciliation protocol. We prove here security against coherent attacks of a reverse-reconciliation protocol under similar assumptions but allowing distances of over 16 km for experimentally feasible parameters. We further clarify the limitations when using the uncertainty relation with quantum memories in security proofs of continuous-variable quantum key distribution.

Figure 1

The diagram shows the measurement setup of Bob's test $\mathcal{T}(\alpha ,T)$. He mixes the incoming signal with a vacuum mode $a$ using a beam splitter with very low reflectivity $1-T$ and applies a heterodyne detection on the reflected signal $a^{\prime }$. The test then consists of checking whether the absolute value of the outcome of the amplitude measurement of mode $t^1$ and the phase measurement on $t^2$ is smaller than $\alpha$.

Figure 2

The plot shows the key rate $\ell /N_{t}ot$ for squeezing and antisqueezing of 11 dB and 16 dB and reconciliation efficiency $\beta =0.95$ depending on the number of signals $N_{t}ot$. Bob's total losses ${\eta }_B$ are $0.45$ (solid line), $0.50$ (dashed line), and $0.55$ (dash-dotted line). Since the source is assumed to be in Alice's laboratory her losses are set to ${\eta }_A=0$. We set the excess noise ${\eta }_{e}x=0.01$, the security parameters to ${\epsilon }_s={\epsilon }_c={10}^{-9}$, and the test parameters to $T=0.99$ and $\alpha =28$.

Figure 3

The plot shows the key rate $\ell /N_{t}ot$ for squeezing and antisqueezing of 11 dB and 16 dB and reconciliation efficiency $\beta =0.90$ depending on the number of signals $N_{t}ot$. Bob's total losses ${\eta }_B$ are $0.40$ (solid line), $0.45$ (dashed line), and $0.50$ (dash-dotted line). The other parameters are as in Fig. 2.

Figure 4

The key rate is plotted against the distance for $N_{t}ot={10}^9$, squeezing and antisqueezing of 11 dB and 16 dB and reconciliation efficiency $\beta$ of $0.95$ (solid line), $0.90$ (dashed line), and $0.85$ (dash-dotted line). We assumed losses of $0.20$ dB per km plus $0.05$ coupling losses. All the other parameters are as in Fig. 2.

Figure 5

The gap between the left-hand side and right-hand side of the uncertainty relations in Eq. (65) (solid line) and (63) (dashed line) are plotted for a two-mode squeezed state with squeezing and antisqueezing of 11 dB and 16 dB against the losses on Bob's mode. The losses on Alice's mode and the excess noise are set to 0. The gap for (65) (solid line) is the amount by which the bound on the key rate reduces in the asymptotic limit compared with the optimal key rate.

Figure 6

The loss dependence of the finite-key rate for $N_{t}ot={10}^{11}$ (straight line) is compared with the asymptotic key rate $r_{\text{UR}}$ (dashed), the optimal key rate $r_{\text{Opt}}$ (dashed-dotted), and the asymptotic key rate for direct reconciliation $r_{\text{DR}}$ (dotted). The squeezing and antisqueezing is set to 11 dB and 16 dB and Alice's coupling losses as well as the excess noise to 0. The reconciliation efficiency of the nonasymptotic key rate is $\beta =0.95$ and the other parameters are as in Fig. 2.