# Error Performance of Amplitude Shift Keying-Type Asymmetric Quantum Communication Systems
## 5 Error Performance
### Performance Comparison with Universal Lower Bound  
Finally, we consider a performance comparison with the universal lower bound on the error probability. For example, when the average number of photons is small, the AQC system using the TSVS almost reaches the universal lower bound, as illustrated in Figure 4c. However, as Figure 4d shows, the gap between the lower bound on the error probability for the TSVS and the universal lower bound increases when the average number of photons increases. This differs from the results reported in reference [9], which stated that when the \(m\)-shot EPR state is used, the universal lower bound is almost always reached, and this is considered to be a characteristic unique to the one-shot pulse case. The characteristics of Figure 4b,f,h become more outstanding and the gap between the lower bound on the error probability for the TSVS and the universal lower bound increases rapidly as the average number of photons increases. In addition, Figure 4b shows that gap between the error probability for the AQC system using the quasi-Bell state and the universal lower bound may also increase rapidly in the same manner as the TSVS. However, Figure 4f shows that the error probability of the AQC system using the quasi-Bell state differs from that of the system using the TSVS in that the gap with respect to the universal lower bound only increases slowly as the average number of photons increases. Furthermore, as shown in Figure 4h, the error probability for the AQC system using the quasi-Bell state almost reaches the universal lower bound even when the average number of photons increases in the case where both \(R_{0}\) and \(R_{1}\) are small.  
To see these details of this characteristics in greater detail, Figure 5 shows the error performance when \(\{R_{0},R_{1}\}=\{0,0.08\}\), \(\{0,0.03\}\) in addition to that when \(\{R_{0},R_{1}\}=\{0,0.3\}\). As shown in Figure 5, \(R_{0}\) is fixed at 0, and as \(R_{1}\) decreases, the error probability for the AQC system using the quasi-Bell state becomes closer to the universal lower bound. To clearly demonstrate the trend of the gap between the error probability when using the quasi-Bell state and the universal lower bound, Figure 6 shows the ratio \(P_{\rm U-LB}/P_{\rm e}^{(\rm Max)}\) with respect to \(R_{1}\) when fixing \(R_{0}\) at 0. (Note the special case where the maximum quasi-Bell state becomes a Bell state when \(\langle n\rangle_{\rm Max}=0.5\)[13].) As is evident in Figure 6, the ratio approaches 1 as \(R_{1}\) decreases in spite of increasing \(\langle n\rangle\). This performance characteristic makes it possible to use the quasi-Bell state to almost reach the universal lower bound in the AQC system, even if severe attenuation--where energy attenuation in the channel can be considered to be included in \(R_{1}\)--occurs in an ultra-long distance channel. (If energy attenuation associated with energy transmissivity \(\eta\) occurs in the channel, then just substitute \(\eta R_{\rm b}\) into \(R_{\rm b}\) in the results. For an example, Figure 4c expresses the error performance when \(R_{0}=0.3\), \(R_{1}=0.6\), and \(\eta=1\); and when \(R_{0}=0.5\), \(R_{1}=0.75\), and \(\eta=0.8\).)  
Figure 5: Error probabilities with respect to the average number of photons \(\langle n\rangle\) when using the maximum quasi-Bell state (denoted by Max Quasi-Bell), the non-maximum quasi-Bell state (denoted by NonMax Quasi-Bell), and the two-mode squeezed vacuum state (denoted by TSVS) with reflectivities of \(\{R_{0},R_{1}\}=\{0,0.3\},\{0,0.08\},\{0,0.03\}\). \(\langle n\rangle\) ranges up to 100. Universal LB and TSVS LB represent the universal lower bound on the error probability and the lower bound on the error probability for TSVS, respectively.