Quantum parameter estimation of a generalized Pauli channel

We present a quantum parameter estimation theory for a generalized Pauli channel Gamma(theta) : S(C-d) --> S(C-d), where the parameter theta is regarded as a coordinate system of the probability simplex Pd2 - 1 We show that for each degree n of extension (id circle times Gamma(theta))(circle timesn) : S((C-d circle times C-d)(circle timesn)) --> S((C-d circle times C-d)(circle timesn)), 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 tau(ME) is an element of S(C-d circle times C-d). We further prove that for the corresponding quantum Cramer-Rao inequality, there is an efficient estimator if and only if the parameter theta is del(m)-affine in Pd2 - 1. These results rely on the fact that the family {id circle times Gamma(theta)(tau(ME))}(theta) of output states can be identified with Pd2 - 1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.