Fig. 1. Schematic diagram of the entanglement-based (EB) model of the four-state modulation protocol using a zero-photon catalysis

Fig. 2. Both and as a function of the modulation variance V.

Fig. 3. Success probability of implementing such a zero-photon catalysis as a function of the transmittance T for several different V. The dashed lines from bottom to top correspond to V = 1.2, 1.3, 1.4, 1.5, respectively.

Fig. 4. Comparison of the performances between the original protocol and the ZPC-based four-state modulation protocol: (a) At a fixed , the secret key rate as a function of the transmission distance with different V = 1.2, 1.3, 1.4, when optimized over the transmittance T; (b) the transmittance T as a function of the transmission distance corresponding to panel (a).

Fig. 5. Comparison of the performances between the original protocol and the ZPC-based four-state modulation protocol: (a) At a fixed , the secret key rate as a function of the transmission distance with different , when optimized over the transmittance T; (b) the transmittance T as a function of the transmission distance corresponding to panel (a).

Fig. 6. Comparison of the performances between the original protocol and the ZPC-based four-state modulation protocol: (a) At a fixed , the secret key rate as a function of the transmission distance with different , when optimized over the transmittance T; (b) the transmittance T as a function of the transmission distance corresponding to panel (a).

Fig. 7. At a fixed , the tolerable excess noise between the original protocol and the ZPC-based four-state modulation protocol as a function of a transmission distance for several different , when optimized over T.