Recently quantum stream cipher based on quantum detection theory so called Yuen-2000 protocol has been developed in both theory and experiment. There are two types of implementation of Y-00 scheme. One is by phase shift keying (PSK), and another is by amplitude or intensity shift keying (ASK). One of the evaluation methods for such stream cipher is a success probability by quantum optimum receiver. In the case of PSK, the signal states have a property of symmetric group, and the quantum limitation has been calculated by our group. In that case, the optimum detection becomes so called square root measurement. Since we are developing Y-00 quantum stream cipher by ASK, we want to clarify the limitation of the attacker in the case of ASK. But in this case, it has no property as group theory, so it is at present difficult to obtain the optimum solution. In this letter, we show the performance of a square root measurement for ASK as sub-optimum receiver and discuss an evaluation of the Y-00 protocol by ASK system.