Thermalization and Return to Equilibrium on Finite Quantum Lattice Systems: Supplementary Material

To lower bound the effective dimension for local Hamiltonian models, we will use a theorem from [1, 2] as a stepping stone. This is a quantum version of the BerryEsseen theorem. The Berry-Esseen theorem is a more powerful statement than the central limit theorem, as it gives the rate of convergence of a distribution to a Gaussian. Let H = ∑ ν Eν |ν〉〈ν| be k-local, i.e., let H be of the form H = ∑ i hi with hi acting only on sites j with dist(i, j) ≤ k. Let ρ have exponentially decaying correlations as in Def. 1. Then, by Lemma 8 of Ref. [1]