For efficient use of energy in quantum communication, it is best to use discrete signals that approach the classical capacity achieved by a continuous input of coherent states. Analytical treatment of the classical communication channel requires discretization, and among such discrete signals, using signals as close as possible to the quantum channel capacity is ideal. The class of binary phase shift keying (BPSK) coherent-state signals has been proven to be optimal in the case of binary discretization; however, for multiary discretization with three or more signals, the optimal signals are unknown. In this study, we consider the case of ternary discretization. We numerically show that 3PSK is the optimal signal constellation in two dimensions.

In recent years, significant progress has been made in the research on errors in quantum deletion/insertion channels along with the development of error-correcting codes. However, the current channel model may not be applicable in specific scenarios, such as when using coherent-state qubits as funda-mental qubits. Here, we materialize the conventional error model of synchronization errors to accommodate coherent-state qubits and their superposition states. Additionally, we quantitatively assess the discrepancies between the conventional error model and our more realistic approach, underscoring the limitations of the former in practical applications.