Disturbance of operation in quantum estimation for the Gaussian P-function

For the quantum Gaussian state family, Hayashi proposed a quantum mechanical operation using beam splitters to estimate the location and scale parameters of the P-function, and he showed that it is asymptotically optimal. In this paper, we analyze the effect of disturbance of his operation caused by the randomness of the transparency of the beam splitters. It is shown that even if the variance of the random transparency is small, Hayashi's estimators are improper in a sense that they are biased and asymptotically inconsistent. In such a case, we propose to stop the operation and correct the biases of estimators.