Universal energy equalization under Haar-random unitary operations

We investigate the work done on finite quantum systems initially in an energy-diagonal ensemble when subjected to Haar-random unitary operations. On average, the system's energy shifts toward the infinite-temperature value, effectively equalizing the energy distribution. This suggests that states with energy above the infinite-temperature value serve as a resource for work extraction. This trend becomes typical in large systems as fluctuations vanish. Numerical simulations for a spin model illustrate this behavior. Similar trends arise from unitary quenches when the quench Hamiltonian satisfies the Eigenstate Thermalization Hypothesis together with additional spectral conditions such as nondegeneracy and flatness, showing that these properties are sufficient to reproduce Haar-typical energy equalization.