We present a quantum parameter estimation theory for a generalized Pauli channel Γ θ : S(C d ) → S(C d ), where the parameter θ is regarded as a coordinate system of the probability simplex P d 2 −1 . We show that for each degree n of extension , the SLD Fisher information matrix for the output states takes the maximum when the input state is an n-tensor product of a maximally entangled state τ M E ∈ S(C d ⊗ C d ). We further prove that for the corresponding quantum Cramér-Rao inequality, there

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... is an efficient estimator if and only if the parameter θ is ∇ m -affine in P d 2 −1 . These results rely on the fact that the family {id ⊗ Γ θ (τ M E )} θ of output states can be identified with P d 2 −1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.