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Quantum Systems Equilibrate Rapidly for Most Observables

Considering any Hamiltonian, any initial state, and measurements with a small number of possible outcomes compared to the dimension, we show that most measurements are already equilibrated. To investigate non-trivial equilibration we therefore consider a restricted set of measurements. When the initial state is spread over many energy levels, and we consider the set of observables for which this state is an eigenstate, most observables are initially out of equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements, where one of the projectors is of low rank, equilibrate rapidly.