Quantum Extremal Surfaces in Holography

• Quantum extremal surfaces are codimension-2 bulk surfaces that extremize the quantum-corrected generalized entropy, bridging quantum information theory and semiclassical gravity.
• They enforce causal constraints and stability in holography through refined variational principles and quantum maximin constructions, ensuring proper entanglement wedge reconstruction.
• In black hole evaporation, QES underpin the formation of islands and the Page curve, offering a resolution to the information paradox with precise quantum corrections.

A quantum extremal surface (QES) is a codimension-2 bulk surface that extremizes the quantum-corrected generalized entropy, playing a central role in semiclassical analysis of holographic entanglement entropy, black hole information, and the structure of entanglement wedges in gauge/gravity duality. The QES prescription extends and supersedes the classical Ryu–Takayanagi (RT) and Hubeny–Rangamani–Takayanagi (HRT) formulas by incorporating bulk quantum fields, von Neumann entropy, and higher-curvature corrections, constituting a nontrivial bridge between quantum information theory and semiclassical gravity (Engelhardt et al., 2014, Mahajan, 4 Feb 2025, Akers et al., 2019).

1. Definition and Variational Principle

A quantum extremal surface, denoted $\mathcal{X}_R$ for a given boundary region $R$, extremizes the “generalized entropy” functional,

$$S_{\mathrm{gen}}[\chi] = \frac{\mathrm{Area}[\chi]}{4G_N} + S_{\mathrm{bulk}}(\Sigma_\chi),$$

where:

• $\chi$ is a codimension-2 bulk surface homologous to $R$,
• $\Sigma_\chi$ is a Cauchy slice bounded by $\chi \cup R$,
• $S_{\mathrm{bulk}}$ is the renormalized von Neumann entropy of quantum fields in $\Sigma_\chi$.

The QES $\mathcal{X}_R$ satisfies $\delta S_{\mathrm{gen}}/\delta\chi = 0$ for all normal deformations $\delta\chi$; among such extremal surfaces, the global minimum is selected (Engelhardt et al., 2014, Mahajan, 4 Feb 2025).

This variational problem generalizes the classical extremality condition $K_a = 0$ (mean curvature vanishing) to

$$K_a + 4G_N \frac{\delta S_{\mathrm{bulk}}}{\delta \chi^a} = 0,$$

where $K_a$ is the mean extrinsic curvature, and the quantum term is the “shape derivative” of the bulk entropy (Engelhardt et al., 2019, Akers et al., 2019).

2. Properties and Uniqueness

Causal Constraints and Barriers

Quantum extremal surfaces always lie outside the bulk causal wedge of the boundary region $R$. Barrier theorems show that if the generalized entropy is everywhere nonincreasing along a null surface $M$, no QES continuously deformable from outside $M$ can cross $M$ (Engelhardt et al., 2014). The quantum focusing conjecture and generalized second law ensure the entanglement wedge, bounded by the QES, is the maximal reconstructible region consistent with boundary causality and subregion duality (Soni, 2024, Akers et al., 2019).

Stability and Classification

The local stability of a QES is controlled by the Hessian of $S_{\mathrm{gen}}$ under normal deformations. Positive-definite Hessian directions correspond to local “throats,” single negative directions to “bulges,” and two or more negative directions to “bounces”—all relevant for the refined covariant analysis of nested or timelike-separated QES configurations (Engelhardt et al., 2023, Akers et al., 2019).

3. Quantum Maximin and Covariant Construction

The quantum maximin construction generalizes the maximin definition of HRT surfaces: on each bulk Cauchy slice containing $R$, find the global minimum of $S_{\mathrm{gen}}$ among surfaces with $\partial\chi = \partial R$; then maximize $S_{\mathrm{gen}}$ over slices. Under mild regularity conditions, this procedure locates the minimal QES for $R$ (Akers et al., 2019).

This result ensures entanglement wedge nesting (the monotonicity of entanglement wedges under nested boundary regions) and strong subadditivity for boundary entropies computed via the QES prescription.

4. Quantum Extremal Surfaces in Black Hole Evaporation and Islands

For evaporating AdS or JT black holes, the QES prescription underpins the “Page curve” for the radiation entropy, resolving the information paradox. As time evolves:

• Early times: no-island saddle, QES at the horizon, entropy grows linearly.
• Late times: QES moves into the black hole interior, the “island” forms, and the entropy saturates to twice the black hole entropy, matching the unitary Page curve (Mahajan, 4 Feb 2025, Hollowood et al., 2021, Pedraza et al., 2021).

Dynamically, the QES can traverse causal structures—e.g., plunging through the horizon, emerging from and re-entering the black hole in JT gravity as radiated intervals change (Hollowood et al., 2021).

In tensor network models, the global minimum of a discrete generalized entropy sharply determines the position of the QES behind the horizon, corresponding to the onset of non-isometry and encoding the transition of information from the black hole to the radiation (Bueller et al., 2024).

5. Refined and One-Shot Quantum Extremal Surface Prescriptions

Recent developments demonstrate that the naive QES prescription is insufficient when the spectrum of bulk states is not “flat” or involves strong one-shot fluctuations. The correct selection rule compares gaps $\Delta A/(4G_N)$ between candidate surfaces to smooth min- and max-entropies $H^\varepsilon_{\min}(b'|b)$ and $H^\varepsilon_{\max}(b'|b)$ of the bulk matter between them:

• If $H^\varepsilon_{\max}(b'|b) < \Delta A/(4G_N)$, the “inner” QES dominates.
• If $H^\varepsilon_{\min}(b'|b) > \Delta A/(4G_N)$, the “outer” QES dominates.
• Otherwise, the true entropy interpolates and cannot be computed from a single extremal surface (Akers et al., 2020, Wang, 2021).

This refined prescription emerges from a replica calculation using the asymptotic equipartition property (AEP), linking holographic entropy transitions to information-theoretic thresholds in quantum state merging and operational one-shot tasks.

6. Bulk Interpretation, Edge Modes, and Kinematic Formulations

In AdS$_3$ gravity, the area term in $S_{\rm gen}$ acquires a canonical interpretation as the entanglement entropy of quantum-group (e.g., $SL_q^+(2,\mathbb{R})$) gravitational edge modes living on the cut surface, revealed by an “open-channel” Euclidean path-integral analysis. These edge modes provide the microstate counting underlying the universal $\mathrm{Area}/(4G_N)$ entropy term, unifying modular-invariance, Gibbons–Hawking, and TQFT perspectives (Wong, 2022).

The geometric data of QES and associated mutual/conditional information can be encoded in the kinematic space formulation: entanglement and information inequalities are encoded as geometric inclusion or convexity properties of regions in a (u,v)-plane equipped with a positive 2-form derived from Crofton’s formula (Gong et al., 2023).

7. Advanced Topics: Modular Curvature, Nonlocality, and Covariant Complexity

Quantum extremal modular curvature (QEMC) characterizes the geometric information encoded in boundary modular transport when islands are present. In the classical (OPE or large-$c$) limit, QEMC locally probes the bulk Riemann curvature near the QES. Away from this limit, QEMC is nonlocal, reflecting the intricate modular entanglement structure associated with islands (Aalsma et al., 2024).

Covariant complexity prescriptions (“Python’s Lunch”) involve non-minimal QES configurations (e.g., bulges, bounces) possibly separated by timelike intervals, signaling that the gravitational tensor-network analogue is inherently covariant—crossing slices as needed to capture all relevant surfaces for bulk decoding problems (Engelhardt et al., 2023).

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Selected Key Equations:

Concept	Formula (in LaTeX)	Reference
Generalized entropy	$$S_{\mathrm{gen}}[\chi] = \frac{\mathrm{Area}[\chi]}{4G_N} + S_{\mathrm{bulk}}(\Sigma_\chi)$$	(Engelhardt et al., 2014)
QES extremality	$$K_a + 4G_N \frac{\delta S_{\mathrm{bulk}}}{\delta \chi^a} = 0$$	(Engelhardt et al., 2019)
Refined QES selection	$$\frac{A_1}{4G} + H^\varepsilon_{\max}(b'|b) < \frac{A_2}{4G}$$ (inner regime), $$\frac{A_1}{4G} + H^\varepsilon_{\min}(b'|b) > \frac{A_2}{4G}$$ (outer regime)	(Akers et al., 2020)
Covariant complexity	$$\log C(p) = \min_{\gamma_0} \max_{\gamma_1 \subset W_O[\gamma_0]} [S_{\mathrm{gen}}(\text{maximinimax}(\gamma_0, \gamma_1)) - S_{\mathrm{gen}}(\gamma_1)]$$	(Engelhardt et al., 2023)
Entropy in kinematic space	$S(A) = \frac{1}{4} \int_{R_A} \omega(u,v)$	(Gong et al., 2023)

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Quantum extremal surfaces are the correct, quantum-corrected holographic duals of fine-grained boundary entropy. They constitute the backbone of modern analyses of holographic entropy inequalities, bulk information recovery, black hole evaporation, and islands, and cannot be replaced by classical extremal or minimal surfaces even at leading semiclassical order (Engelhardt et al., 2014, Akers et al., 2019, Akers et al., 2020, Wong, 2022, Mahajan, 4 Feb 2025).