  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 varphi_{\theta_{0}+ u/\sqrt{n}}^{n} consisting of joint states of n
identically prepared quantum systems approaches in a statistical sense a family
of Gaussian state phi_{u} of an algebra of canonical commutation relations. The
convergence holds for all "local parameters" u\in R^{m} such that
theta=theta_{0}+ u/sqrt{n} parametrizes a neighborhood of a fixed point
theta_{0}\in Theta\subset R^{m}.
  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 investigate the relation
between the two notions of convergence. For reader's convenience and
completeness we review the relevant results of the classical as well as the
quantum theory.
