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.

翻译：量子技术日益依赖于专门的统计推断方法来分析量子测量数据，这推动了“量子统计学”的发展——该领域在量子物理学与“经典”统计学的交叉中逐渐成形。迄今为止，研究较少的课题之一是无限维量子系统的统计推断，其可视为非参数统计学的量子对应。本文分析了由纯态等量制备的量子系综构成的量子统计模型的渐近理论。在系综尺寸趋近无穷的极限下，我们建立了该独立同分布模型与量子高斯白噪声模型的局部渐近等价性。利用这一局部渐近等价性结果，我们确定了隶属Hermite-Sobolev波函数类的纯态估计的最小最大速率。此外，针对同一类态的二次泛函估计，我们观察到速率中的“肘效应”现象；而在非参数Hermite-Sobolev类中对纯态进行假设检验时，则可达到尖锐的参数速率。