The problem of specifying the optimum receiver in quantum detection theory is considered for application to optical communications. The quantum minimax rule is formulated and a necessary and sufficient condition for it is given. This rule provides a powerful calculation method in obtaining the quantum optimum solution. Several examples are worked out to prove superiority of the minimax strategy in practical problems. For the case of ternary coherent signals, in particular, is obtained an explicit equation of error probability. Furthermore the minimax rule gives a good penetrating calculation method to obtain suboptimum receivers in case accompanied by an additive external noise.

The problems for the controllability of the quantum state are considered for application to optical communications. The new optimization problems arise in the quantum communication theory in addition to problems considered up to now. The concrete example of the controllability problems is Two-photon coherent state (TCS) generated by Two-photon laser. The optimum conditions for the parameters of TCS, which minimize the error probability, are given when TCS is employed as the transmission state in the quantum optimum receiver of the binary system, and its numerical behavior is shown. Furthermore the noise behavior associated with the TCS in the several optical receivers are shown. As the result, the ultimate noise limit hitherto can be reduced by using TCS in the optical communication systems. The saving power, when we employ the TCS as the transmission state in the binary communication system, is about 3 dB in the direct photo detector, and it is 5 dB in the optical homodyne receiver. Furthermore the saving power in the quantum optimum receiver is 4 dB. The SNR of the direct photo detector is improved by more than 6 dB. The information performance of the homo dyne is superior to the heterodyne when TCS is employed, while the heterodyne is superior to the homodyne in general when an ordinary coherent state is employed. Thus, it is suggested for reduction of the noise problems that the Two-photon laser which will be realized in the future will convey the significant benefits to the optical communications.