Relative Entropy of Random States and Black Holes

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 fmd 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.

Automated summary

The study focuses on the relative entropy of highly excited quantum states, where it is found that analytical results precisely match small-N numerics and random matrix theory can accurately capture the behavior of chaotic many-body eigenstates. When applied to the AdS/CFT correspondence, it is observed that black hole microstates remain distinguishable even with limited access to the quantum state, albeit with a nonperturbatively small distinguishability in Newton's constant. These results are interpreted within the context of the subsystem eigenstate thermalization hypothesis (SETH), indicating that holographic systems follow SETH up to subsystems half the size of the total system.