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Information-theoretic ideas have provided numerous insights in the progressof fundamental physics, especially in our pursuit of quantum gravity. Inparticular, the holographic entanglement entropy is a very useful tool instudying AdS/CFT, and its efficacy is manifested in the recent black hole pagecurve calculation. On the other hand, the one-shot information-theoreticentropies, such as the smooth min/max-entropies, are less discussed in AdS/CFT.They are however more fundamental entropy measures from the quantum informationperspective and should also play pivotal roles in holography. We combine thetechnical methods from both quantum information and quantum gravity to put thisidea on firm grounds. In particular, we study the quantum extremal surface(QES) prescription that was recently revised to highlight the significance ofone-shot entropies in characterizing the QES phase transition. Motivated by theasymptotic equipartition property (AEP), we derive the refined quantum extremalsurface prescription for fixed-area states via a novel AEP replica trick,demonstrating the synergy between quantum information and quantum gravity. Wefurther prove that, when restricted to pure bulk marginal states, suchcorrections do not occur for the higher R\'enyi entropies of a boundarysubregion in fixed-area states, meaning they always have sharp QES transitions.Our path integral derivation suggests that the refinement applies beyondAdS/CFT, and we confirm it in a black hole toy model by showing that the Pagecurve, for a black hole in a superposition of two radiation stages, receives alarge correction that is consistent with the refined QES prescription.

The refined quantum extremal surface prescription from the asymptotic equipartition property