Entanglement-ergodic quantum systems equilibrate exponentially well. (arXiv:1802.02052v1 [quant-ph])

One of the outstanding problems in non-equilibrium physics is to precisely
understand when and how physically relevant observables in many-body systems
equilibrate under unitary time evolution. While general equilibration results
have been proven that show that equilibration is generic provided that the
initial state has overlap with sufficiently many energy levels, at the same
time results showing that natural initial states fulfill this condition are
lacking. In this work, we present stringent results for equilibration for
ergodic systems in which the amount of entanglement in energy eigenstates with
finite energy density grows volume-like with the system size. Concretely, we
carefully formalize notions of entanglement-ergodicity in terms of R\'enyi
entropies, from which we derive that such systems equilibrate exponentially
well. Our proof uses insights about R\'enyi entropies and combines them with
recent results about the probability distribution of energy in lattice systems
with initial states that are weakly correlated.

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