# Error Performance of Amplitude Shift Keying-Type Asymmetric Quantum Communication Systems
## 6 Conclusions  
In this paper, we have proposed an ASK-type AQC system as a step toward development of a new asymmetric communication system. In this AQC system, Bob, who has a high transmission capability, transmits one of the entangled light beams, which acts as a communication medium, to Alice, who has a low transmission capability; Alice operates on the light beam to encode the information that she wants to send to Bob, and then sends the light beam back to Bob. Bob then decodes the information received from Alice by performing an optimum quantum measurement (i.e., a joint measurement) of the other entangled light beam and the light beam that returned from Alice. As a first step toward evaluation of the system performance, we focused on the communication performance from Alice to Bob, and investigated the basic performance characteristics based on the error probability criterion. First, using the quasi-Bell state as the light source, we derived an analytical expression for the error probability by using an \(8\times 8\) matrix representation to express the density operators of the two received quantum states affected by the reflectivities \(\{R_{0},R_{1}\}\), which corresponded to the binary information that Alice wants to send. Then, using this analytical expression, we compared the superior performances of the AQC systems using the TSVS, the maximum quasi-Bell state, and the non-maximum quasi-Bell state with that of the asymmetric classical communication (ACC) system in terms of their error probabilities. As a result, it was clarified that the error probabilities of the AQC systems using the maximum and non-maximum quasi-Bell states differed only slightly. In addition, the error probability of the AQC system using the quasi-Bell state is always lower than that of the ACC system, regardless of the reflectivity setting, and the AQC system using the quasi-Bell state also shows a clear performance advantage over the system using the TSVS when a sufficiently large average number of photons is used. In fact, as described in Section 2, when the average number of photons is large, the amount of entanglement of the TSVS is overwhelmingly greater than that of the quasi-Bell state, but the results in this paper show that the performance of the AQC system using the quasi-Bell state is overwhelmingly better than that of the same system using the TSVS. Therefore, the performance of the AQC system should be determined by selecting a type of entangled state that is suitable for the system, rather than by considering the amount of entanglement. The conclusion above is strengthened by the fact that the performance of the ACC system surpassed that of the AQC system using the TSVS when the average number of photons was sufficiently large. What causes quasi-Bell state to work better than the TSVS? Unfortunately, as far as we know, there are no studies in terms of the comparison between the quasi-Bell state and the TSVS, except for references [18; 19], although the reason for the superiority of the quasi-Bell state and some related potential properties of that have not been elucidated. These studies also reveal the fact that there  
Figure 6: Ratio of the universal lower bound on the error probability to the error probability when using the maximum quasi-Bell state with respect to reflectivity \(R_{1}\), where \(R_{0}\) is fixed at 0.  
are some entanglement-based systems or protocols that require a suitable entangled state rather than a large amount of entanglement to improve its performance. Additionally, we believe the advantage of the quasi-Bell state over the TSVS comes from the robustness of coherent states against attenuation. However, a perfect explanation is not yet available. We consider the issue to be a very challenging topic that has value as a future subject. Getting back to the main topic, when \(R_{0}\) is small or when \(R_{1}-R_{0}\) is large, the AQC system using the quasi-Bell states shows a clear advantage when only a small average number of photons is used.