A Berry-Esseen Bound for Quantum Lattice Systems

TL;DR

This paper proves a Berry-Esseen bound for quantum lattice systems, providing a quantitative estimate of how local observable measurements converge to a normal distribution as system size increases.

Contribution

It establishes a rigorous convergence rate towards the normal distribution for local observables in quantum lattice systems, extending the central limit theorem with explicit error bounds.

Findings

Measurement of local observables follows a normal distribution with error scaling as N^{-1/2} polylog(N).

The result applies to systems with finite correlation length and local Hamiltonians.

The convergence estimate is optimal up to logarithmic factors.

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.