  We derive the fluctuation theorem for quantum-state statistics that can be
obtained when we initially measure the total energy of a quantum system at
thermal equilibrium, let the system evolve unitarily, and record the
quantum-state data reconstructed at the end of the process. The obtained
theorem shows that the quantum-state statistics for the forward and backward
processes is related to the equilibrium free-energy difference through an
infinite series of independent relations, which gives the quantum work
fluctuation theorem as a special case, and reproduces the out-of-time-order
fluctuation-dissipation theorem near thermal equilibrium. The quantum-state
statistics exhibits a system-size scaling behavior that differs between
integrable and non-integrable (quantum chaotic) systems as demonstrated
numerically for one-dimensional quantum lattice models.
