Leading order corrections to the quantum extremal surface prescription

TL;DR

The paper addresses paradoxes that arise when applying the naïve quantum extremal surface (QES) prescription in holography in the presence of multiple competing surfaces and significant bulk entropy. It introduces leading-order refinements based on one-shot information theory, replacing von Neumann conditional entropy with smooth min-/max-entropies and deriving refined QES conditions that depend on $H^{\varepsilon}_{\max}(b'|b)$ and $H^{\varepsilon}_{\min}(b'|b)$ relative to $(A_2-A_1)/(4G)$; these refinements are justified first in fixed-area states via a careful replica-trick analysis and then extended to general holographic states (via RTNs and averaging). A key outcome is that entanglement wedge reconstruction (EWR) becomes state-specific and is best understood as one-shot quantum state merging, possibly employing zero-bits, with a precise operational interpretation through the Petz map. The framework naturally generalizes to configurations with more than two extremal surfaces, introducing min-EW and max-EW and a set of inequalities that determine when the naive QES suggests wrong leading-order physics. Overall, the work provides a coherent, operationally grounded refinement of holographic entropy and boundary reconstruction, with implications for Page curves, black hole information, and bit-thread pictures.

Abstract

We show that a naïve application of the quantum extremal surface (QES) prescription can lead to paradoxical results and must be corrected at leading order. The corrections arise when there is a second QES (with strictly larger generalized entropy at leading order than the minimal QES), together with a large amount of highly incompressible bulk entropy between the two surfaces. We trace the source of the corrections to a failure of the assumptions used in the replica trick derivation of the QES prescription, and show that a more careful derivation correctly computes the corrections. Using tools from one-shot quantum Shannon theory (smooth min- and max-entropies), we generalize these results to a set of refined conditions that determine whether the QES prescription holds. We find similar refinements to the conditions needed for entanglement wedge reconstruction (EWR), and show how EWR can be reinterpreted as the task of one-shot quantum state merging (using zero-bits rather than classical bits), a task gravity is able to achieve optimally efficiently.

Figures (9)

• Figure 1: Setup in which we derive a contradiction from a naïve application of the QES prescription. The boundary is divided into two subregions, $B$ and $\overline{B}$. For both, there are two competing quantum extremal surfaces, $\gamma_1$ and $\gamma_2$, with $\gamma_1$ homotopic to $\overline{B}$ and $\gamma_2$ to $B$. We take $B$ to be larger, such that the area of $\gamma_2$ is bigger than that of $\gamma_1$ at $\mathcal{O}(1)$. Between these surfaces is a large amount of matter (the "dustball"), such that some states of the matter have entropy much larger than the difference in areas of the two surfaces.
• Figure 2: Comparison of the naïve QES entropy to the correct, "refined" answer, for the state \ref{['eq:mixture_state']} in the setup of Figure \ref{['fig:dustball']}. While the slope of the refined answer is controlled by $(A_2 - A_1)/4G$, the slope of the naïve answer is controlled by $S_\mathrm{thermal} > (A_2-A_1)/4G$. The naïve answer is in general larger than the refined one by an $\mathcal{O}(1/G)$ amount.
• Figure 3: Two entangled dustballs. Like Figure \ref{['fig:dustball']}, but now we consider the entropy of $\overline{B}R$, where $R$ is an entire extra copy of the boundary, dual to its own dustball. The two dustballs are in a mixture of entangled states, given by \ref{['eq:ent_dustball_state']}. A naïve application of the QES prescription gives the wrong answer for the entropy $S(\overline{B} R)$.
• Figure 4: Black hole setup in which we derive a contradiction from the QES prescription. Practically identical to the setup of Figure \ref{['fig:dustball']}, this setup replaces the dustball with a black hole, and its boundary regions $B$ and $\overline{B}$ are now connected.
• Figure 5: Eigenvalue density for the three Regimes in Section \ref{['sec:example1']}. In Regime 1, there are two peaks of eigenvalues, each associated to one of the two states in the mixture, and each much greater than the critical value $1/e^{A_2/4G}$. Hence the naïve QES prescription is correct. In Regime 2, one of the peaks has shifted to the critical value, while the other remained where it was, leading to large corrections in the naïve QES prescription. In Regime 3, both peaks have moved to the critical value, and the naïve prescription is valid again. Note the agreement with numerical results for the analogous random tensor network in Appendix \ref{['app:numerics']}.

Theorems & Definitions (8)

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