A Berry-Esseen theorem is proven for local observables in quantum lattice systems with finite correlation length, yielding convergence to normality with error O(N^{-1/2} polylog N).
Esseen,On the Liapounoff Limit of Error in the Theory of Probability, Arkiv för matematik, astronomi och fysik (Almqvist & Wiksell, 1942)
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A Berry-Esseen Bound for Quantum Lattice Systems
A Berry-Esseen theorem is proven for local observables in quantum lattice systems with finite correlation length, yielding convergence to normality with error O(N^{-1/2} polylog N).