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.

There is a gap between properties of quantum and classical reliability functions. That is, the upper bound of the quantum reliability function diverges, whereas that of classical one does not, even if the same coherent signal is used. We are interested in the origin of this gap. In this paper, we investigate the upper bound of the reliability function for quantum signals coded by simplex codes when using square-root measurement (SRM). The reliability function in this case has a much steeper slope with respect to the rate than in the uncoded case. This result suggests the divergence of the upper bound of the quantum reliability function in the previous study.

In quantum communication theory, group covariant signals are important and useful when we consider the minimization problem of the error probability or the maximization problem of the mutual information. However, in practical applications, there are important non-symmetric signals such as amplitude shift keying or quadrature amplitude modulation signals. To manage this situation, we have proposed a symmetrization method of non-symmetric pure-state signals by using coding. In this paper, we consider to apply the symmetrization method to mixed-state signals and give a necessary condition of mixed-state signals to apply the method. Then, we present examples of mixed-state signals to which the symmetrization method can be applied.