Geometric Constraints from Subregion Duality Beyond the Classical Regime

TL;DR

The paper investigates how subregion duality in AdS/CFT constrains bulk geometry beyond the classical limit, focusing on Entanglement Wedge Nesting (EWN) and Causal wedge containment ($\mathcal{C}(A)\subseteq \mathcal{E}(A)$) and their connections to energy/entropy inequalities. It shows a chain of implications where EWN implies $\mathcal{C}\subseteq \mathcal{E}$, which implies the Boundary Causality Condition (BCC), and where $\mathcal{C}\subseteq \mathcal{E}$ implies boundary QHANEC, while EWN implies boundary QNEC, with all results holding to all orders in $G\hbar$ via the quantum extremal surface (QES) framework. The work also uses Wall's Lemma and analyzes loop/higher-derivative corrections, showing that the leading structure is preserved and clarifying the role of generalized entropy $S_{\rm gen}=\frac{A}{4G\hbar}+S_{\rm out}+Q$. Overall, the paper strengthens the consistency of subregion duality and holographic entanglement entropy in the quantum regime and outlines paths for generalization beyond AdS and for deeper connections to field-theoretic proofs of energy conditions.

Abstract

Subregion duality in AdS/CFT implies certain constraints on the geometry: entanglement wedges must contain causal wedges, and nested boundary regions must have nested entanglement wedges. We elucidate the logical connections between these statements and the Quantum Focussing Conjecture, Quantum Null Energy Condition, Boundary Causality Condition, and Averaged Null Energy Condition. Our analysis does not rely on the classical limit of bulk physics, but instead works to all orders in $G\hbar \sim 1/N$. This constitutes a nontrivial check on the consistency of subregion duality, entanglement wedge reconstruction, and holographic entanglement entropy beyond the classical regime.

Figures (5)

• Figure 1: The logical relationships between the constraints discussed in this paper. The left column contains semi-classical quantum gravity statements in the bulk. The middle column is composed of constraints on bulk geometry. In the right column is quantum field theory constraints on the boundary CFT. All implications are true to all orders in $G\hbar \sim 1/N$. We have used dashed implication signs for those that were proven to all orders before this paper.
• Figure 2: The causal relationship between $e(A)$ and $D(A)$ is pictured in an example spacetime that violates $\mathcal{C} \subseteq \mathcal{E}$. The boundary of $A$'s entanglement wedge is shaded. Notably, in $\mathcal{C}\subseteq \mathcal{E}$ violating spacetimes, there is necessarily a portion of $D(A)$ that is timelike related to $e(A)$. Extremal surfaces of boundary regions from this portion of $D(A)$ are necessarily timelike related to $e(A)$, which violates EWN.
• Figure 3: The boundary of a BCC-violating spacetime is depicted, which gives rise to a violation of $\mathcal{C} \subseteq \mathcal{E}$. The points $p$ and $q$ are connected by a null geodesic through the bulk. The boundary of $p$'s lightcone with respect to the AdS boundary causal structure is depicted with solid black lines. Part of the boundary of $q$'s lightcone is shown with dashed lines. The disconnected region $A$ is defined to have part of its boundary in the timelike future of $q$ while also satisfying $p \in D(A)$. It follows that $e(A)$ will be timelike related to $D(A)$ through the bulk, violating $\mathcal{C} \subseteq \mathcal{E}$.
• Figure 4: The surface $M$ and $N$ are shown touching at a point $p$. In this case, $\theta_M < \theta_N$. The arrows illustrate the projection of the null orthogonal vectors onto the Cauchy surface.
• Figure 5: This picture shows the various vectors defined in the proof. It depicts a cross-section of the extremal surface at constant $z$. $e(A)_{vac}$ denotes the extremal surface in the vacuum. For flat cuts of a null plane on the boundary, they agree. For wiggly cuts, they will differ by some multiple of $k^i$.