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December 2018 Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise
Cristina Butucea, Mădălin Guţă, Michael Nussbaum
Ann. Statist. 46(6B): 3676-3706 (December 2018). DOI: 10.1214/17-AOS1672


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.


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Cristina Butucea. Mădălin Guţă. Michael Nussbaum. "Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise." Ann. Statist. 46 (6B) 3676 - 3706, December 2018.


Received: 1 May 2017; Revised: 1 December 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 06965701
MathSciNet: MR3852665
Digital Object Identifier: 10.1214/17-AOS1672

Primary: 62B15
Secondary: 62G05, 62G10, 81P50

Rights: Copyright © 2018 Institute of Mathematical Statistics


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Vol.46 • No. 6B • December 2018
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