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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 Γθ specified by a parameter θ. It is shown that the quantum Fisher information of the family of output states maximized over all input states , which quantifies the ultimate statistical distinguishability of the parameter θ, 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 ⊗ Γθ)⊗n to be O(n) is derived. This criterion further proves that for almost all quantum channels, the maximum quantum Fisher information increases in the order of O(n).
Fujiwara et al. (2008) studied this question.