A fibre bundle over manifolds of quantum channels and its application to quantum statistics

A fibre bundle structure is introduced over manifolds of quantum channels. This structure has a close connection with the problem of estimating an unknown quantum channel Gamma(theta) specified by a parameter theta. It is shown that the quantum Fisher information of the family of output states (id circle times Gamma(theta))((sigma) over tilde) maximized over all input states (sigma) over tilde, which quantifies the ultimate statistical distinguishability of the parameter theta, is expressed in terms of a geometrical quantity on the fibre bundle. Using this formula, a criterion for the maximum quantum Fisher information of the nth extended channel (id circle times Gamma(theta))(circle times n) to be O(n) is derived. This criterion further proves that for almost all quantum channels, themaximum quantum Fisher information increases in the order of O(n).