LOCAL ASYMPTOTIC EQUIVALENCE OF PURE STATES ENSEMBLES AND QUANTUM GAUSSIAN WHITE NOISE

• 成果类型: Article
• 署名作者: Butucea, Cristina; Guta, Madalin; Nussbaum, Michael
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; Ecole Polytechnique; Universite Paris Saclay; University of Nottingham; Cornell University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1672
• 发表日期: 2018
• 页码: 3676-3706
• 关键词: Adaptive Estimation; tomography; entropy; Minimax

摘要

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 nonparametric 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.