A Lyapunov type central limit theorem for mixing quantum systems

Physical systems, composed of interacting identical (or similar) subsystems appear in many branches of physics. They are standard in condensed matter physics. Assuming that each subsystem only interacts with its nearest neighbours and that the energy per subsystem has an upper limit, which must not depend on the number of subsystems n, we show that the distribution of energy eigenvalues of almost every product state converges to the Gaussian normal distribution in the limit of infinitely many subsystems. To the best of our knowledge, this fundamental quantum feature has not yet been recognized in the literature [1, 2, 3]. Central limit theorems for the distribution of energy eigenvalues in quantum gases with Boltzmann statistics [4] as well as for Bose and Fermi statistics [5] have been discussed by M. Sh. Goldstein. His theorems apply for mixed states, namely classical mixtures of quantum states involving classical probabilities. We consider here the distribution of energy eigenvalues for a pure quantum state. In our case, the distribution is thus of purely quantum nature, it solely exists because the state we consider is not an eigenstate of the energy operator of our system. Our theorem may be viewed as a central limit theorem for mixing quantum systems.