Two quantum analogues of Fisher information from a large deviation viewpoint of quantum estimation

We discuss two quantum analogues of the Fisher information, the symmetric logarithmic derivative (SLD) Fisher information and Kubo-Mori-Bogoljubov (KMB) Fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of superefficient estimators. It is shown that the difference between them is characterized by the change of the order of limits.