The Annals of Statistics

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

Cristina Butucea, Mădălin Guţă, and Michael Nussbaum

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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.

Article information

Ann. Statist., Volume 46, Number 6B (2018), 3676-3706.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62B15: Theory of statistical experiments
Secondary: 62G05: Estimation 62G10: Hypothesis testing 81P50: Quantum state estimation, approximate cloning

Le Cam distance local asymptotic equivalence quantum Gaussian process quantum Gaussian sequence quantum states ensemble nonparametric estimation quadratic functionals nonparametric sharp testing rates


Butucea, Cristina; Guţă, Mădălin; Nussbaum, Michael. Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise. Ann. Statist. 46 (2018), no. 6B, 3676--3706. doi:10.1214/17-AOS1672.

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Supplemental materials

  • Supplement to “Local asymptotic equivalence of pure states ensembles and quantum Gaussian white noise”. A more detailed overview of asymptotic equivalence for classical models is provided in Appendix A.1. The results on quadratic functionals and nonparametric testing are further discussed in Appendix A.2 and A.3. Proofs of all results are given in Appendix B.