  This is an expanded version of the short report [Phys. Rev. Lett. 126, 171603
(2021)], where the relative entropy was used to distinguish random states drawn
from the Wishart ensemble as well as black hole microstates. In this work, we
expand these ideas by computing many generalizations including the Petz R\'enyi
relative entropy, sandwiched R\'enyi relative entropy, fidelities, and trace
distances. These generalized quantities are able to teach us about new
structures in the space of random states and black hole microstates where the
von Neumann and relative entropies were insufficient. We further generalize to
generic random tensor networks where new phenomena arise due to the locality in
the networks. These phenomena sharpen the relationship between holographic
states and random tensor networks. We discuss the implications of our results
on the black hole information problem using replica wormholes, specifically the
state dependence (hair) in Hawking radiation. Understanding the differences
between Hawking radiation of distinct evaporating black holes is an important
piece of the information problem that was not addressed by entropy calculations
using the island formula. We interpret our results in the language of quantum
hypothesis testing and the subsystem eigenstate thermalization hypothesis
(ETH), deriving that chaotic (including holographic) systems obey subsystem ETH
for all subsystems less than half the total system size.
