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arXiv · 2605.03829

A Berry-Esseen Bound for Quantum Lattice Systems

Marcus Cramer·Fernando G. S. L. Brandão·Mădălin Guţă·Álvaro M. Alhambra·Matteo Scandi
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Abstract

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 $N$ 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 $\mathcal{O}\left(N^{-\frac{1}{2}} \text{polylog}(N)\right)$, which is optimal up to logarithmic factors.

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Marcus Cramer, Fernando G. S. L. Brandão, Mădălin Guţă, Álvaro M. Alhambra, Matteo Scandi. 2026-05-05. A Berry-Esseen Bound for Quantum Lattice Systems. https://arxiv.org/abs/2605.03829

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