Non-Gaussian postselection and virtual photon subtraction in continuous-variable quantum key distribution

Photon subtraction can enhance the performance of continuous-variable quantum key distribution (CV QKD). However, the enhancement effect will be reduced by the imperfections of practical devices, especially the limited efficiency of a single-photon detector. In this paper, we propose a non-Gaussian postselection method to emulate the photon substraction used in coherent-state CV QKD protocols. The virtual photon subtraction not only can avoid the complexity and imperfections of a practical photon-subtraction operation, which extends the secure transmission distance as the ideal case does, but also can be adjusted flexibly according to the channel parameters to optimize the performance. Furthermore, our preliminary tests on the information reconciliation suggest that in the low signal-to-noise ratio regime, the performance of reconciliating the postselected non-Gaussian data is better than that of the Gaussian data, which implies the feasibility of implementing this method practically.

Figure 1

(a) Entanglement-based (EB) scheme of CV QKD with photon subtraction. (b) Prepare-and-measure (PM) scheme of CV QKD with equivalent postselection as virtual photon subtraction. Het: heterodyne detection; Hom: homodyne detection; BS1(2): beam splitter; $\gamma$: Alice's measurement result; $\lambda$: parameter of TMSV; $Q\left(\gamma ,\lambda ,T\right)$: postselection filter function; $T\left(\eta \right)$: transmittance of BS1(2); $T_C,\varepsilon$: channel parameters.

Figure 2

(a) The maximal secret key rate at each transmission distance, when changing the transmittance $T$ of Alice's BS1. (b) The optimal $T$ for the maximal secret key rate in (a). The uppermost black solid line in (a) represents the case of original protocol. Other lines represent one-photon subtraction (blue solid line), two-photon subtraction (green dashed line), three-photon subtraction (pink dotted line), and four-photon subtraction (red dash-dotted line), respectively. The simulation parameters are as follows: the variance of TMSV state is $V=20$, channel loss is $a=0.2\mathrm{dB}$/km, excess noise is $\varepsilon =0.01$, and reconciliation efficiency is $\beta =0.95$.

Figure 3

(a) The maximal tolerable excess noise at each transmission distance, when changing the transmittance $T$ of Alice's BS1. (b) The optimal $T$ for the maximal tolerable excess noise in (a). The uppermost black solid line in (a) represents the case of original protocol. Other lines represent one-photon subtraction (blue solid line), two-photon subtraction (green dashed line), three-photon subtraction (pink dotted line), and four-photon subtraction (red dash-dotted line), respectively. The simulation parameters are $V=20,a=0.2\mathrm{dB}$/km, $\varepsilon =0.01$, and $\beta =0.95$.

Figure 4

Secret key rate vs transmission distance and transmittance $T$ of Alice's BS1. The middle black solid line represents the optimal $T$ of each distance, when the secret key rate approaches its optimal value ($R_{\mathrm{opt}}$). The left red dash-dotted (dashed) line represents the lower bound of $T$ for each distance, when its secret key rate is 90% (50%) of its optimum at that distance. The right blue dash-dotted (dashed) line represents the upper bound of $T$ for each distance, when its secret key rate is 90% (50%) of its optimum at that distance. The simulation parameters are $V=20,a=0.2\mathrm{dB}$/km, $\varepsilon =0.01$, and $\beta =0.95$.

Figure 5

The detection efficiency of SPD will influence the performance of the scheme using photon subtraction. The lines from top to bottom are as follows: the original protocol without photon subtraction (black solid line), with one-photon subtraction under unit DE (blue solid line), 0.8 DE (green dashed line), 0.5 DE (pink dotted line), and with on-off detector under 0.1 DE (red dash-dotted line), respectively. The simulation parameters are the variance $V=20$, channel loss $a=0.2\mathrm{dB}$/km, excess noise $\varepsilon =0.01$, transmittance of $\mathrm{BS}1 T=0.8$, and reconciliation efficiency $\beta =0.95$.

Figure 6

The success probability of subtracting $k$ photons of a TMSV with different transmittances $T$ of Alice's BS. The lines from top to bottom represent one-photon subtraction (blue solid line), two-photon subtraction (green dashed line), three-photon subtraction (pink dotted line), and four-photon subtraction (red dash-dotted line), respectively. The variance of TMSV is $V=20$, and Alice uses ideal SPD.