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Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise

Butucea, Cristina; Guţă, Mădălin; Nussbaum, Michael

Authors

Cristina Butucea

Michael Nussbaum



Abstract

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.

Citation

Butucea, C., Guţă, M., & Nussbaum, M. (2018). Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise. Annals of Statistics, 46(6B), 3676-3706. https://doi.org/10.1214/17-aos1672

Journal Article Type Article
Acceptance Date Dec 8, 2017
Online Publication Date Dec 20, 2018
Publication Date Dec 20, 2018
Deposit Date Jan 3, 2019
Publicly Available Date Jan 3, 2019
Journal The Annals of Statistics
Print ISSN 0090-5364
Publisher Institute of Mathematical Statistics (IMS)
Peer Reviewed Peer Reviewed
Volume 46
Issue 6B
Pages 3676-3706
DOI https://doi.org/10.1214/17-aos1672
Keywords Le Cam distance, local asymptotic equivalence, quantum Gaussian process, quantum Gaussian sequence, quantum states ensemble, nonparametric estimation, quadratic functionals, nonparametric sharp testing rates
Public URL https://nottingham-repository.worktribe.com/output/1441415
Publisher URL https://projecteuclid.org/euclid.aos/1536631287

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