High-bit-rate continuous-variable quantum key distribution

Here, we demonstrate that a practical continuous-variable quantum key distribution (QKD) protocol relying on the Gaussian modulation of coherent states features secret key rates that cannot be achieved with standard qubit discrete-variable QKD protocols. Notably, we report a practical postprocessing that allows us to extract more than 1 bit of secret key per channel use.

Figure 1

Schematic representation of the slice protocol for direct reconciliation. First the input source is quantized into an $m$-bit source. Then each of the $m$ sources is encoded and sent to Bob. In the figure the first slice is transmitted unencoded. The decoder takes as side information its own source and with the $m$ encoded sources produces an estimate of the quantized source.

Figure 2

Two examples of quantizers dividing the real line in $2^3$ intervals. The figure on the top shows a quantizer with constant step; the figure on the bottom shows a quantizer with optimized interval length.

Figure 3

Evolution of the value of the constant step giving the best quantization efficiency with respect to the SNR. The solid red line corresponds to 3 slices (8 intervals), the dashed green line corresponds to 4 slices (16 intervals), and the dotted blue line corresponds to 5 slices (32 intervals).

Figure 4

Optimal quantization efficiency with respect to the SNR for a discretization of the real line into regular intervals (solid lines) and nonregular intervals (dashed lines). Red lines give the discretization efficiency for 3 slices (8 intervals), green lines give it for 4 slices (16 intervals), and blue lines give it for 5 slices (32 intervals).

Figure 5

Evolution of the value of the optimal rates for each slice with respect to the SNR for an optimal discretization of the real line into regular intervals. The lowest plots correspond to the least significant bits, which are the noisiest bits. Solid red lines correspond to three slices, dashed green lines correspond to four slices, and dotted blue lines correspond to five slices.

Figure 6

Factor $\gamma$ as defined in Eq. (4) indicating the sensitivity of slice reconciliation to the suboptimality of the error correcting codes used for three, four, and five slices.

Figure 7

Overall efficiency of slice reconciliation with error correcting codes of efficiency ${\beta }_c=95\%$ for three, four, and five slices.

Figure 8

Comparison of the RR (dashed) and DR (solid) secret key rate with ideal measurement devices. The thick lines show secret key rate, the thin lines the optimal SNR which is equal to $V_A/T$ up to some small excess noise related term. The secret key rate of the first four km is zoomed in the upper right corner. Parameters: $\alpha =0.2,\xi =0.0015V_A$.

Figure 9

Secret key rate in the DR scenario. From top to bottom the curves show the secret key rate with: ideal measurement devices (solid), realistic devices characterized by a finite detection efficiency $\eta$ and an electronic noise $v_{\mathrm{elec}}$ (dashed), slice reconciliation (dotted) and reconciliation over a BIAWGN (dashed-dotted). The thick lines show secret key rate, the thin lines the optimal SNR which is a function of the input signal variance. Parameters: $\xi =0.0015V_A,\alpha =0.2,\eta =0.6,v_{\mathrm{elec}}=0.01$.

Figure 10

Secret key rate in the RR scenario. From top to bottom the curves show the secret key rate with: ideal measurement devices (solid), realistic devices characterized by a finite detection efficiency $\eta$ and an electronic noise $v_{\mathrm{elec}}$ (dashed), slice reconciliation (dotted) and reconciliation over a BIAWGN (dashed-dotted). The thick lines show secret key rate, the thin lines the optimal SNR which is a function of the input signal variance. The secret key rate of the first twenty km is zoomed in at the upper right corner. Parameters: $\xi =0.0015V_A,\alpha =0.2,\eta =0.6,v_{\mathrm{elec}}=0.01$.

Figure 11

Secret key rate in the RR scenario with a heterodyne detection and the security proof of [25]. The solid lines show the asymptotic secret key rate, the dashed lines show the secret key rate with finite blocks of length $n={10}^9$. From top to bottom the upper solid and upper dashed curves show the secret key rate for realistic devices characterized by a detection efficiency $\eta =0.85$ and the lower solid and lower dashed curves for a detection efficiency $\eta =0.6$. We consider here the paranoid mode where the noise added by the detection can be manipulated by the attacker. Other parameters: $\xi =0.0015V_A,\alpha =0.2$, block size $n={10}^9$, electronic noise $v_{\mathrm{elec}}=0.001$ and security parameter $\epsilon ={10}^{-10}$.

Figure 12

Secret key rate in the RR scenario with a heterodyne detection and the security proof of [25]. The figure shows the secret key rate with respect to the detection efficiency. From top to bottom the upper curve shows the secret key rate in the asymptotic regime and the lower curve shows the secret key in the finite key scenario. We consider here the paranoid mode where the noise added by the detection can be manipulated by the attacker. Other parameters: electronic noise $v_{\mathrm{elec}}=0.001$, distance $d=0.1\mathrm{km},\xi =0.0015V_A,\alpha =0.2$, block size $n={10}^{10}$ and security parameter $\epsilon ={10}^{-10}$.