Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime

TL;DR

The paper extends holographic entanglement entropy beyond the classical regime by introducing quantum extremal surfaces that extremize the generalized entropy $S_ ext{gen}$. It shows that, at leading quantum order, this framework reproduces FLM and aligns with the HRT prescription in the classical limit, while predicting deeper-probing surfaces that lie outside the causal wedge. It proves several results: quantum extremal surfaces cannot be reached by causal signals, and barrier theorems limit bulk reconstruction, implying fundamental limits on what boundary data can reveal about the bulk. These insights reshape our understanding of bulk reconstruction and suggest intriguing directions for higher-order corrections and nonperturbative quantum gravity.

Abstract

We propose that holographic entanglement entropy can be calculated at arbitrary orders in the bulk Planck constant using the concept of a "quantum extremal surface": a surface which extremizes the generalized entropy, i.e. the sum of area and bulk entanglement entropy. At leading order in bulk quantum corrections, our proposal agrees with the formula of Faulkner, Lewkowycz, and Maldacena, which was derived only at this order; beyond leading order corrections, the two conjectures diverge. Quantum extremal surfaces lie outside the causal domain of influence of the boundary region as well as its complement, and in some spacetimes there are barriers preventing them from entering certain regions. We comment on the implications for bulk reconstruction.

Figures (6)

• Figure 1: A surface $E$ splits an AdS-Cauchy slice $\Sigma$ into Ext$(E)$ and Int$(E)$. The generalized entropy can be defined with respect to either side, depending on whether we pick $S_{in}$ or $S_{out}$ to calculate $S_{\text{gen}}$. In the case where the state is pure, the choices yield identical results.
• Figure 2: The dashed line represents a future causal horizon $H^{+}$, the solid green lines are regions within the horizon and the solid blue lines are outside it. $C$ is an infinitesimal region on $H^{+}$, along which the entropies $S_\mathrm{out}$ or $S_\mathrm{in}$ are evolved. Strong subadditivity says that $S(AC)-S(A)\geq S(B)-S(BC)$, so that $S_\mathrm{in}$ increases faster than $S_\mathrm{out}$.
• Figure 3: (a) $\mathcal{T}$ is a quantum trapped surface with infinitesimal normal $\vec{\delta \mathcal{T}}$, and $N$ is the null surface generated from null congruences shot from $\mathcal{T}$. We could consider variations of $S_{\mathrm{gen}}$ in any direction $\delta \vec{T}$, but we take them only in the direction $\vec{k}$, which is a null generator of $N$. $\mathcal{T}$ is quantum trapped, so we know that the generalized entropy on $\mathcal{T}$ along $\vec{k}$ must be strictly decreasing. (b) There are two possible directions in which one could deform a codimension 2 surface. $\delta T^{a}$ is defined as any combination of the two.
• Figure 4: The causal surface $C_R$ is the spacelike boundary of the causal wedge (depicted in green) associated with $R$. The domain of dependence $D_{R}$ of $R$ is depicted in purple. The quantum extremal surface $\mathcal{X}_{R}$ does not intersect the causal wedge.
• Figure 5: A projection of $H^{+}_{R}$ (depicted in red) and $H^{+}_{\Xi}$ (depicted in blue) onto a slice $\Sigma$ passing through $\mathcal{X}_{R}$ (dashed black).

Theorems & Definitions (13)

• Theorem 2.1
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 5.1
• proof
• Theorem 5.2