On the distribution of the mean energy in the unitary orbit of quantum states

Given a closed quantum system, the states that can be reached with a cyclic process are those with the same spectrum as the initial state. Here we prove that, under a very general assumption on the Hamiltonian, the distribution of the mean extractable work is very close to a gaussian with respect to the Haar measure. We derive bounds for both the moments of the distribution of the mean energy of the state and for its characteristic function, showing that the discrepancy with the normal distribution is increasingly suppressed for large dimensions of the system Hilbert space.

Automated summary

This research proves that, under certain conditions, the energy distribution of states reached through cyclic processes closely resembles a Gaussian distribution with respect to the Haar measure. Limits are derived for the average energy of the state, indicating that the deviation from the normal distribution diminishes as the dimension of the system's Hilbert space increases.