  A pure quantum state can fully describe thermal equilibrium as long as one
focuses on local observables. Thermodynamic entropy can also be recovered as
the entanglement entropy of small subsystems. When the size of the subsystem
increases, however, quantum correlations break the correspondence and cause a
correction to this simple volume-law. To elucidate the size dependence of the
entanglement entropy is of essential importance in linking quantum physics with
thermodynamics, and in addressing recent experiments in ultra-cold atoms. Here
we derive an analytic formula of the entanglement entropy for a class of pure
states called cTPQ states representing thermal equilibrium. We further find
that our formula applies universally to any sufficiently scrambled pure states
representing thermal equilibrium, i.e., general energy eigenstates of
non-integrable models and states after quantum quenches. Our universal formula
can be exploited as a diagnostic of chaotic systems; we can distinguish
integrable models from chaotic ones and detect many-body localization with high
accuracy.
