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

Butucea, Cristina; Gu??, M?d?lin; Nussbaum, Michael

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., Gu??, 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