The refined quantum extremal surface prescription from the asymptotic equipartition property

Jinzhao Wang

Institute for Theoretical Physics, ETH 8093 Zürich, Switzerland

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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.

Information-theoretic ideas are more powerful than ever in helping us understand fundamental physics. In particular, the entanglement entropy has been a useful probe of correlations in quantum field theory and gravity. Its importance culminates in holography, where the entanglement entropy can be “geometrized” via the Ryu-Takayanagi formula, and the so inspired quantum extremal surface (QES) prescription recently serves as the key to solving the black hole information paradox. In quantum information theory, on the other hand, the von Neumann entropy is operationally understood as an “average” quantity, which only becomes relevant in the asymptotic regime where the law of large number kicks in. While it is considered a fundamental quantity in the gravity picture, it does not have this fundamental role in information theory. Instead, we think of the one-shot entropies as fundamental measures of information.

Motivated by the mismatch, we propose a novel replica trick, which is a method used to compute the entanglement entropy in field theory. Our novel replica trick can potentially be useful in other situations where the standard replica method faces the difficulty of analytic continuation. We demonstrate its efficacy in refining the phase transition mechanism responsible for resolving the black hole information paradox, manifesting the significance of the one-shot entropies. In particular, our path-integral derivation implies a revised Page curve that characterizes the entropy of the Hawking radiation. This correction is also confirmed in our work, showing the universal applicability of the refined QES prescription in gravity.

We combine both the ideas and techniques from quantum information and quantum gravity to propose a novel replica trick to compute entanglement entropies. We then derive a refined phase transition mechanism responsible for the black hole Page curve. We think both the method and result are of general interest to the physics community, and it is indeed a truly interdisciplinary work showing the synergy between quantum information and quantum gravity.

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Cited by

[1] Jonah Kudler-Flam and Pratik Rath, "Large and small corrections to the JLMS Formula from replica wormholes", Journal of High Energy Physics 2022 8, 189 (2022).

[2] Christopher Akers and Geoff Penington, "Quantum minimal surfaces from quantum error correction", SciPost Physics 12 5, 157 (2022).

[3] Renato Renner and Jinzhao Wang, "The black hole information puzzle and the quantum de Finetti theorem", arXiv:2110.14653.

[4] Keiichiro Furuya, Nima Lashkari, and Mudassir Moosa, "Renormalization group and approximate error correction", arXiv:2112.05099.

The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 20:59:23) and SAO/NASA ADS (last updated successfully 2022-10-04 20:59:24). The list may be incomplete as not all publishers provide suitable and complete citation data.