# Error Performance of Amplitude Shift Keying-Type Asymmetric Quantum Communication Systems
## 5 Error Performance
### Performance Comparison of Use of Maximum and Non-Maximum Quasi-Bell States  
Figure 2 plots the error probabilities for the AQC system versus the reflectivities \(R_{0}\) and \(R_{1}\) (\(R_{0}<R_{1}\)), which are changed from 0 to 1 when the transmitted average number of photons in the maximum and non-maximum quasi-Bell states is fixed at \(\langle n\rangle_{\text{Max}}=\langle n\rangle_{\text{NonMax}}=1\). The error probability when the maximum quasi-Bell state is used is represented by the green curved surface, and the error probability when the non-maximum quasi-Bell state is used is represented by the orange curved surface.  
Figure 2 shows that the error probability \(P_{\text{e}}^{\text{(Max)}}\) when the maximum quasi-Bell state is used tends to decrease when \(R_{1}-R_{0}\) increases. In particular, the figure shows reasonable results that confirm that the maximum error probability is achieved when \(R_{0}\approx R_{1}\), and that the minimum error probability is achieved when \(R_{0}=0\) and \(R_{1}=1\). Comparison of the error probabilities of the AQC system when using the maximum and non-maximum  
quasi-Bell states shows that the maximum quasi-Bell state is superior in most cases, but there is no significant difference between the two cases. Furthermore, use of the maximum quasi-Bell state does not always provide superior results, because the AQC system using the non-maximum quasi-Bell state is superior to the corresponding system using the maximum quasi-Bell state in the extreme case in which \(R_{1}-R_{0}\) is very small and \(R_{0}\approx 1\) or \(R_{0}\approx 0\). The details can be seen in Figure 3 regarding the contour plot of the ratio, \(P_{\rm e}^{\rm(NonMax)}/P_{\rm e}^{\rm(Max)}\), of the error probability when using the non-maximum state to that when using the maximum quasi-Bell state. Figure 3 shows that the ratio is between 1 and 1.1 in most cases, but the ratio may be less than 1 in the extreme case.  
Actually, there are several studies comparing the performance between the maximum and non-maximum quasi-Bell states, such as quantum teleportation [15], quantum super-dense coding [34], quantum reading [33], and quantum illumination [18; 19]. Reference [15] showed that the non-maximum quasi-Bell state offers an advantage over the maximum quasi-Bell state at small coherent amplitudes, and may offer more resistance to attenuation than the maximum quasi-Bell state. References [18; 19; 33; 34] showed that the non-maximum quasi-Bell state offers more resistance to attenuation and phase noise than the maximum quasi-Bell state, in some special cases, and the result of this paper is one more piece of evidence in terms of that. That evidence also reveals a fact that the maximum quasi-Bell state offers better performance than the non-maximum quasi-Bell state in ideal cases, such as cases with large coherent amplitudes or environments without noise, but the opposite may be true in some unideal cases. Therefore, for some applications using the quasi-Bell state at small coherent amplitudes or environments with noise, the non-maximum quasi-Bell state which offers more resistance to noise may play an important role. We must be careful not to  
Figure 3: Contour plot of the ratio of the error probability when using the non-maximum quasi-Bell state (\(P_{\rm e}^{\rm(NonMax)}\)) to that when using the maximum quasi-Bell state (\(P_{\rm e}^{\rm(Max)}\)). The average number of photons is fixed at 1.  
Figure 2: Error probability characteristics with respect to the reflectivities \(\{R_{0},R_{1}\}\) when using the maximum quasi-Bell state (green curved surface) and the non-maximum quasi-Bell state (orange curved surface). The average number of photons is fixed at 1.  
dismiss the value of using the non-maximum quasi-Bell state, although the reason in terms of its superiority has not yet been elucidated. We consider the issue to be an interesting topic that has value as a future subject.