Continuous-variable measurement-device-independent quantum key distribution with virtual photon subtraction

The method of improving the performance of continuous-variable quantum key distribution protocols by postselection has been recently proposed and verified. In continuous-variable measurement-device-independent quantum key distribution (CV-MDI QKD) protocols, the measurement results are obtained from untrusted third party Charlie. There is still not an effective method of improving CV-MDI QKD by the postselection with untrusted measurement. We propose a method to improve the performance of coherent-state CV-MDI QKD protocol by virtual photon subtraction via non-Gaussian postselection. The non-Gaussian postselection of transmitted data is equivalent to an ideal photon subtraction on the two-mode squeezed vacuum state, which is favorable to enhance the performance of CV-MDI QKD. In CV-MDI QKD protocol with non-Gaussian postselection, two users select their own data independently. We demonstrate that the optimal performance of the renovated CV-MDI QKD protocol is obtained with the transmitted data only selected by Alice. By setting appropriate parameters of the virtual photon subtraction, the secret key rate and tolerable excess noise are both improved at long transmission distance. The method provides an effective optimization scheme for the application of CV-MDI QKD protocols.

Figure 1

Entanglement-based scheme of CV-MDI QKD with virtual photon subtraction. EPR is the two-mode squeezed vacuum state. All green detectors represent homodyne detection. PNR is the photon number resolving detector. $T_A$ ($T_B$) is the channel transmittance from Alice (Bob) to Charlie. D is the displacement operation of Bob. QM is the quantum memory.

Figure 2

(a) The maximal secret key rate with optimal $T_{PS}^A$ in (b) by setting different $k$ vs PLOB bound of equivalent photon subtraction only in Alice's side. (b) The maximal tolerable noise with optimal $T_{PS}^A$ by setting different $k$ of equivalent photon subtraction only in Alice's side. In the simulations, the variance of TMSV state is $V_A=V_B=40$, the excess noise is $\epsilon =0.002$, the reconciliation efficiency is $\beta =0.95$.

Figure 3

The maximal secret key rate with optimal $T_{PS}^B$ by setting different $k$ of equivalent photon subtraction only in Bob's side. In the simulations, the variance of TMSV state is $V_A=V_B=40$, the excess noise is $\epsilon =0.002$, the reconciliation efficiency is $\beta =0.95$.

Figure 4

The comparison of maximal secret key rate among the original CV-MDI QKD; the CV-MDI QKD with virtual photon subtraction in only Alice's side, Bob's side, and both their sides. In the simulations, the variance of TMSV state is $V_A=V_B=40$, the excess noise is $\epsilon =0.002$, the reconciliation efficiency is $\beta =0.95$.

Figure 5

The comparison of maximal secret key rate among the original CV-MDI QKD and the CV-MDI QKD with non-Gaussian postselection under two independent attacks and the optimal correlated attacks. In the simulations, the variance of TMSV state is $V_A=V_B=10000$, the excess noise is $\epsilon =0.002$, the reconciliation efficiency is $\beta =0.95$.

Figure 6

Entanglement-based scheme of photon-subtracted TMSV source using an ideal photon number resolving detector. $T_{PS}$ is transmittance of the beam splitter. PNR is the photon number resolving detector.

Figure 7

(a) The success probability $P_S^k$ of virtual photon subtraction by setting different transmittance $T_{PS}$. (b) The optimal $T_{PS}^A$ of virtual photon subtraction for the maximal secret key rate by setting different $k$. In the simulations, the variance of TMSV state is $V_A=V_B=40$, the excess noise is $\epsilon =0.002$, and the reconciliation efficiency is $\beta =0.95$.