Quantum parameter estimation of a generalized Pauli channel

We present a quantum parameter estimation theory for a generalized Pauli channel Γ_θ : (^d) → (^d), where the parameter θ is regarded as a coordinate system of the probability simplex ^d2−1. We show that for each degree n of extension (id ⊗ Γ_θ)^⊗n : ((^d ⊗ ^d)^⊗n) → ((^d ⊗ ^d)^⊗n), 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^ME ∊ (^d ⊗ ^d). We further prove that for the corresponding quantum Cramér–Rao inequality, there is an efficient estimator if and only if the parameter θ is ∇^m-affine in ^d2−1. These results rely on the fact that the family {id ⊗ Γ_θ(τ^ME)}_θ of output states can be identified with ^d2−1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.