# Error Performance of Amplitude Shift Keying-Type Asymmetric Quantum Communication Systems
## 5 Error Performance
### Performance Comparison Using Each Quantum State  
Figure 4a,c,e,g shows the error probabilities obtained when the average number of photons \(\langle n\rangle(:=\langle n\rangle_{\mathrm{U-LB}}=\langle n\rangle_{\mathrm{TSVS }}=\langle n\rangle_{\mathrm{Max}}=\langle n\rangle_{\mathrm{NonMax}}= \langle n\rangle_{\mathrm{Coh}})\), which is regarded as the signal energy, is varied from 0.5 to 5 for the AQC system when using the TSVS, the maximum quasi-Bell state, and the non-maximum quasi-Bell state, and the ACC system using the coherent state, for which the reflectivities \(\{R_{0},R_{1}\}\) were fixed at \(\{0.5,1\},\{0.3,0.6\},\{0.2,0.8\},\) and \(\{0,0.3\}\), respectively. The pink chain line represents the error probability \(P_{\mathrm{e}}^{\mathrm{(Hom)}}\) for the ACC system, and the blue and orange dotted lines represent the error probabilities \(P_{\mathrm{e}}^{\mathrm{(Max)}}\) and \(P_{\mathrm{e}}^{\mathrm{(NonMax)}}\) for the AQC system when using the maximum and non-maximum quasi-Bell states, respectively. The black dashed line represents the error probability \(P_{\mathrm{e}}^{\mathrm{(TSVS)}}\) for the AQC system when using the TSVS, and the green and red dashed lines represent the upper bound \(P_{\mathrm{UB}}^{\mathrm{(TSVS)}}\) and the lower bound \(P_{\mathrm{LB}}^{\mathrm{(TSVS)}}\), respectively, for \(P_{\mathrm{e}}^{\mathrm{(TSVS)}}\). The gray solid line represents the universal lower bound on the error probability, \(P_{\mathrm{U-LB}}\). The horizontal axis represents the average number of photons, and the vertical axis represents the error probability in each case. The minimum average number of photons that can be considered is 0.5 because the minimum average number of photons in the maximum quasi-Bell state is \(\langle n\rangle_{\mathrm{Max}}=0.5\), and cases smaller than that are not defined. If the average number of photons is greater than 5, it then becomes difficult to calculate the error probability \(P_{\mathrm{e}}^{\mathrm{(TSVS)}}\); therefore, only the upper bound \(P_{\mathrm{UB}}^{\mathrm{(TSVS)}}\) and the lower bound \(P_{\mathrm{LB}}^{\mathrm{(TSVS)}}\) are used for the evaluation of this probability. Figure 4b,d,f,h shows the error performances corresponding to those shown in Figure 4a,c,e,g, respectively, when the average number of photons \(\langle n\rangle\) is increased from 0.5 to 100. These figures confirm that the AQC system using the quasi-Bell state always maintains a clear performance advantage over the ACC system, despite increases in the average number of photons \(\langle n\rangle\). Although the error probability of the AQC system when using the quasi-Bell state is similar to that of the ACC system, in that it decreases exponentially in tandem with the increase in \(\langle n\rangle\), the AQC system can achieve the same error probability when using only half of the average number of photons used by the ACC system. Conversely, Figure 4b,d,f,h shows the error probability for the AQC system using the TSVS approaches that of the ACC system with increasing \(\langle n\rangle\). In particular, as shown in Figure 4f, the ACC system provides superior performance to the AQC system using the TSVS when \(\langle n\rangle>35\). Figure 4g,h shows that the performance of the ACC system exceeds that of the AQC system using the TSVS when the average number of photons is smaller, i.e., when \(\langle n\rangle\approx 5\). This is contrary to the results presented in reference [9], which indicated that the performance obtained when using the \(m\)-shot EPR state exceeds that of the coherent state as the average number of photons increases, and this performance is considered to be a characteristic unique to the one-shot pulse case. Consideration of these figures together with Figure 4b,d shows that the ACC system exceeds the performance of the AQC system using the TSVS with smaller average number of photons when \(R_{0}\) is small or when \(R_{1}-R_{0}\) is large.