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).

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This paper explores an entirely new application of the quantum entanglement. The problem treated here is the quantum channel identification problem: given a parametric family {Γ θ } θ of quantum channels, find the best strategy of estimating the true value of the parameter θ. As a simple example, we study the estimation problem of the isotropic depolarization parameter θ for a two level quantum system H ≅ C 2. In the framework of noncommutative statistics, it is shown that the optimal input state on H ⊗ H to the channel exhibits a transition-like behavior according to the value of the parameter θ.

We develop a theory of local asymptotic normality in the quantum domain based on a novel quantum analogue of the log-likelihood ratio. This formulation is applicable to any quantum statistical model satisfying a mild smoothness condition. As an application, we prove the asymptotic achievability of the Holevo bound for the local shift parameter.

It is shown that for an adaptive quantum estimation scheme based on locally unbiased measurements, the sequence of maximum likelihood estimators is strongly consistent and asymptotically efficient.