  Quantum technology is increasingly relying on specialised statistical
inference methods for analysing quantum measurement data. This motivates the
development of "quantum statistics", a field that is shaping up at the overlap
of quantum physics and "classical" statistics. One of the less investigated
topics to date is that of statistical inference for infinite dimensional
quantum systems, which can be seen as quantum counterpart of non-parametric
statistics. In this paper we analyse the asymptotic theory of quantum
statistical models consisting of ensembles of quantum systems which are
identically prepared in a pure state. In the limit of large ensembles we
establish the local asymptotic equivalence (LAE) of this i.i.d. model to a
quantum Gaussian white noise model. We use the LAE result in order to establish
minimax rates for the estimation of pure states belonging to Hermite-Sobolev
classes of wave functions. Moreover, for quadratic functional estimation of the
same states we note an elbow effect in the rates, whereas for testing a pure
state a sharp parametric rate is attained over the nonparametric
Hermite-Sobolev class.
