Quantum parameter estimation of a generalized Pauli channel

Abstract

We present a quantum parameter estimation theory for a generalized Pauli channel Γθ : (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 ⊗ Γθ)⊗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 τME (d ⊗ d). We further prove that for the corresponding quantum Cramer–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 ⊗ Γθ(τME)}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.