A Berry-Esseen Bound for Quantum Lattice Systems

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.