  The entanglement properties of random quantum states or dynamics are
important to the study of a broad spectrum of disciplines of physics, ranging
from quantum information to high energy and many-body physics. This work
investigates the interplay between the degrees of entanglement and randomness
in pure states and unitary channels. We reveal strong connections between
designs (distributions of states or unitaries that match certain moments of the
uniform Haar measure) and generalized entropies (entropic functions that depend
on certain powers of the density operator), by showing that R\'enyi
entanglement entropies averaged over designs of the same order are almost
maximal. This strengthens the celebrated Page's theorem. Moreover, we find that
designs of an order that is logarithmic in the dimension maximize all R\'enyi
entanglement entropies, and so are completely random in terms of the
entanglement spectrum. Our results relate the behaviors of R\'enyi entanglement
entropies to the complexity of scrambling and quantum chaos in terms of the
degree of randomness, and suggest a generalization of the fast scrambling
conjecture.
