ua nt - p h / 06 06 21 3 v 1 2 6 Ju n 20 06 Local asymptotic normality in quantum statistics

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family φ θ0+u/ √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all “local parameters” u ∈ R such that θ = θ0 + u/ √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ R. In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and show that weak and strong convergence are equivalent in the case of finite number of parameters for experiments based on type I algebras with discrete center. For reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory.