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

摘要

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 Θ}, discussed by Amari, is extended to the same manifold but for quantum states (density matrices), S = {(θ);θ 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)