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

## TL;DR

This paper establishes that large ensembles of identically prepared pure quantum states are asymptotically equivalent to a quantum Gaussian white noise model, enabling new statistical inference methods for quantum systems.

## Contribution

It proves the local asymptotic equivalence between i.i.d. pure quantum states and quantum Gaussian white noise, and derives minimax estimation rates for these states.

## Key findings

- LAE holds for pure quantum state ensembles and quantum Gaussian white noise.
- Minimax rates are established for pure state estimation within Hermite-Sobolev classes.
- A sharp parametric rate is achieved for testing pure states in a nonparametric setting.

## 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.