# Generalized entanglement entropies of quantum designs. (arXiv:1709.04313v1 [quant-ph])

The entanglement properties of random quantum states and channels are
important to the study of a broad spectrum of disciplines of physics, ranging
from quantum information to condensed matter to high energy. Ensembles of
quantum states or unitaries that reproduce the first $\alpha$ moments of
completely random states or unitary channels (drawn from the Haar measure) are
called $\alpha$-designs. Entropic functions of the $\alpha$-th power of a
density operator are called $\alpha$-entropies (e.g.~R\'enyi and Tsallis). We
reveal strong connections between the orders of designs and generalized (in
particular R\'enyi) entropies, by showing that the R\'enyi-$\alpha$
entanglement entropies averaged over (approximate) $\alpha$-designs are
generically almost maximal. Moreover, we find that the min entanglement
entropies become maximal for designs of an order logarithmic in the dimension
of the system, which implies that they are indistinguishable from uniformly
random by the entanglement spectrum. Our results relate the complexity of
scrambling to the degree of randomness by R\'enyi entanglement entropy.