Exponential and mixture families in quantum statistics: Dual structure and unbiased parameter estimation

The differential-geometric formulation of statistics (the so-called information geometry) concerning the structure of a smooth manifold in the parameter space Θ of classical probabilities, S = {p(·,θ),θ ϵ Θ}, discussed by Amari, is extended to the same manifold but for quantum states (density matrices), S = {ϱ(θ);θ ϵ Θ} in N × N matrix algebras. This is done by introducing an n-tuple of tangent vectors {δ}ni = 1 in analogy to the classical ones {∂i}ni = 1. On this basis, a special problem of quantum information geometry is treated; namely, the analysis of the exponential and the mixture families defined, respectively, as (e) ϱ(θ) = exp(θi Ai − ψ(θ)). θ ϵ Θ = Rn. Ai ϵ Bs(HN). (m) ϱ(θ) = θiAi + θ0 A0. θ ϵ Θ = (0,1)n + 1. ∑i=0nθi=1. Ai ϵ B+(HN) Tr Ai = 1 (the tensorial summation convention for repeated indices is used). ve some of the basic theorems known in the classical information geometry by extending the formulation to a non-commutative smooth manifold. We establish the existence of a pair of dual affine coordinate systems in (e) or (m) and a projection theorem in order to ensure the Cramer-Rao inequality and an identification of the efficient estimator.