  We study the relative entropy of highly excited quantum states. First, we
sample states from the Wishart ensemble and develop a large-N diagrammatic
technique for the relative entropy. The solution is exactly expressed in terms
of elementary functions. We compare the analytic results to small-N numerics,
finding precise agreement. Furthermore, the random matrix theory results
accurately match the behavior of chaotic many-body eigenstates, a manifestation
of eigenstate thermalization. We apply this formalism to the AdS/CFT
correspondence where the relative entropy measures the distinguishability
between different black hole microstates. We find that black hole microstates
are distinguishable even when the observer has arbitrarily small access to the
quantum state, though the distinguishability is nonperturbatively small in
Newton's constant. Finally, we interpret these results in the context of the
subsystem Eigenstate Thermalization Hypothesis (sETH), concluding that
holographic systems obey sETH up to subsystems half the size of the total
system.
