  Proving thermalization from the unitary evolution of a closed quantum system
is one of the oldest questions that is still nowadays only partially resolved.
Several efforts have led to various formulations of what is called the
eigenstate thermalization hypothesis, which leads to thermalization under
certain conditions on the initial states. These conditions, however, are
sensitive to the precise formulation of the hypothesis. In this work, we focus
on the important case of low entanglement initial states, which are
operationally accessible in many natural physical settings, including
experimental schemes for testing thermalization and for quantum simulation. We
prove thermalization of these states under precise conditions that have
operational significance. More specifically, motivated by arguments of
unavoidable finite resolution, we define a random energy smoothing on local
Hamiltonians that leads to local thermalization when the initial state has low
entanglement. Finally we show that such a transformation affects neither the
Gibbs state locally nor, under generic smoothness conditions on the spectrum,
the short-time dynamics.
