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arXiv:2209.00832 · The Annals of Statistics

Efficiency of estimators for locally asymptotically normal quantum statistical models

Akio Fujiwara, Koichi Yamagata

Published Jun 1, 2023• 4 citations
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Abstract

We herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators, such as quantum regular estimators and quantum minimax estimators, leading to a universal tight lower bound beyond the i.i.d. assumption. This formulation complements the theory of quantum contiguity developed in the previous paper [Fujiwara and Yamagata, Bernoulli 26 (2020) 2105–2141], providing a solid foundation of the theory of weak quantum local asymptotic normality.

DOI
10.1214/23-aos2285

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We herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators, such as quantum regular estimators and quantum minimax estimators, leading to a universal tight lower bound beyond the i.i.d. assumption. This formulation complements the theory of quantum contiguity developed in the previous paper [Fujiwara and Yamagata, Bernoulli 26 (2020) 2105–2141], providing a solid foundation of the theory of weak quantum local asymptotic normality.
Local asymptotic normality is a cornerstone of classical asymptotic statistics. Extending it to quantum statistical models entails formidable conceptual and technical challenges arising from their noncommutative nature, above all, the definition of a quantum likelihood ratio, whose very choice dictates the subsequent theoretical development. Employing quantum information geometry based on the symmetric logarithmic derivative, we formulate a quantum Lebesgue decomposition and the square-root likelihood ratio. These constructions provide a natural framework for a quantum analogue of Le Cam’s third lemma, the formulation of quantum local asymptotic normality, and the asymptotic representation results developed in our earlier work linking regular quantum statistical models to quantum Gaussian shift models. In this way, quantum information geometry emerges as a natural and unifying foundation for asymptotic quantum statistics.
In classical statistics, Pinsker’s theorem provides an exact asymptotic minimax bound in nonparametric estimation, improving upon optimal rates of convergence results. We obtain a quantum version of the theorem by establishing asymptotic minimax results for estimation of the displacement vector in a quantum Gaussian white noise model, given by a sequence of shifted vacuum states. Analogous results are then obtained for estimation of a general pure state from an ensemble of identically prepared, independent quantum systems, using the recently established local asymptotic equivalence to the quantum Gaussian white noise model. Optimality holds with respect to the most fundamental distance measure for quantum states, that is, trace norm distance, and in a true quantum sense, allowing for all possible measurements. Adaptive estimators are also obtained for the above cases. As an application, we obtain asymptotic minimax adaptive estimators for Wigner functions of pure states.
It is expected that the statistical fluctuations of local observables in large quantum systems obey the central limit theorem, and approximate a normal distribution as their size grows. Here, we prove a version of the Berry-Esseen theorem for quantum lattice systems, which strengthens that central limit theorem by providing a rigorous convergence estimate towards the normal distribution for large but finite system size. Given a local quantum Hamiltonian on NNN particles and a quantum state with a finite correlation length, the result states that the measurement of local observables such as the energy follows a normal distribution, up to an error scaling as O(N−12polylog(N))\mathcal{O}\left(N^{-\frac{1}{2}} \text{polylog}(N)\right)O(N−21​polylog(N)), which is optimal up to logarithmic factors.