Abstract
Information-theoretic ideas have provided numerous insights in the progress of fundamental physics, especially in our pursuit of quantum gravity. In particular, the holographic entanglement entropy is a very useful tool in studying AdS/CFT, and its efficacy is manifested in the recent black hole page curve calculation. On the other hand, the one-shot information-theoretic entropies, such as the smooth min/max-entropies, are less discussed in AdS/CFT. They are however more fundamental entropy measures from the quantum information perspective and should also play pivotal roles in holography. We combine the technical methods from both quantum information and quantum gravity to put this idea on firm grounds. In particular, we study the quantum extremal surface (QES) prescription that was recently revised to highlight the significance of one-shot entropies in characterizing the QES phase transition. Motivated by the asymptotic equipartition property (AEP), we derive the refined quantum extremal surface prescription for fixed-area states via a novel AEP replica trick, demonstrating the synergy between quantum information and quantum gravity. We further prove that, when restricted to pure bulk marginal states, such corrections do not occur for the higher Rényi entropies of a boundary subregion in fixed-area states, meaning they always have sharp QES transitions. Our path integral derivation suggests that the refinement applies beyond AdS/CFT, and we confirm it in a black hole toy model by showing that the Page curve, for a black hole in a superposition of two radiation stages, receives a large correction that is consistent with the refined QES prescription.
Highlights
The notion of entanglement entropy is getting popular in our pursuit of understanding fundamental physics
We show its efficacy in the context of AdS/CFT by deriving the recently revised quantum extremal surface (QES) prescription due to Akers and Penington (AP) [79], in analogy with how Lewkowycz–Maldacena derived the Ryu-Takayanagi forumla using the standard replica trick [22, 32]
Our derivation features techniques from both quantum information and quantum gravity, so here we introduce the essential tools for readers who are not familiar with both subjects
Summary
The notion of entanglement (von Neumann) entropy is getting popular in our pursuit of understanding fundamental physics. To properly resolve the transition over the indefinite regime, one needs to add up the contributions from non-replica-symmetric saddles in the gravitational path integral calculation [79,80,81] This can sometimes be done for simple states via the resolvent method [54, 79], which effectively extracts the entanglement spectrum from the integer Renyi entropies, but is hard to implement for general states. Our derivation of the refined QES prescription exactly showcases this advantage, where the resolvent calculation cannot be implemented for general states This very approach of getting to the von Neumann entropy via AEP has been used in various other applications in quantum theory, from proving the strong subadditivity [85] and its strengthened version [86] to entropic uncertainty relations [87]. The technical details that come into the derivation and the calculations are provided in Appendix A and C, and we sketch a complementary max-entropy replica method for obtaining upper bounds in Appendix B
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