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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 drawnfrom the Wishart ensemble as well as black hole microstates. In this work, weexpand these ideas by computing many generalizations including the Petz R\'enyirelative entropy, sandwiched R\'enyi relative entropy, fidelities, and tracedistances. These generalized quantities are able to teach us about newstructures in the space of random states and black hole microstates where thevon Neumann and relative entropies were insufficient. We further generalize togeneric random tensor networks where new phenomena arise due to the locality inthe networks. These phenomena sharpen the relationship between holographicstates and random tensor networks. We discuss the implications of our resultson the black hole information problem using replica wormholes, specifically thestate dependence (hair) in Hawking radiation. Understanding the differencesbetween Hawking radiation of distinct evaporating black holes is an importantpiece of the information problem that was not addressed by entropy calculationsusing the island formula. We interpret our results in the language of quantumhypothesis testing and the subsystem eigenstate thermalization hypothesis(ETH), deriving that chaotic (including holographic) systems obey subsystem ETHfor all subsystems less than half the total system size.

Distinguishing Random and Black Hole Microstates