Quantum Extremal Surface Method

• Quantum Extremal Surface Method is a framework that extends the classical Ryu–Takayanagi formula by incorporating quantum corrections to compute fine-grained entanglement entropy.
• It uses extremization and the maximin construction to select codimension-2 surfaces that minimize a generalized entropy combining geometric area and von Neumann entropy of bulk quantum fields.
• The method underpins key insights into black hole information paradoxes, island formation, and Page curve transitions through refined semiclassical and replica trick analyses.

The quantum extremal surface (QES) method generalizes the classical Ryu–Takayanagi (RT) formula for holographic entanglement entropy to include quantum corrections and to address the computation of fine-grained entropy in semiclassical gravity. The QES is defined as a codimension-2 bulk surface that extremizes the “generalized entropy” functional, which is the sum of the Bekenstein–Hawking area term and the von Neumann entropy of bulk quantum fields across the surface. The QES prescription is central in the consistent semiclassical description of black hole information, the emergence of “islands,” phase transitions such as the Page curve, and in resolving paradoxes in both holographic and non-holographic settings.

1. Generalized Entropy Functional and Quantum Extremality

The QES prescription associates to any boundary subregion $A$ the entropy

$$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$

where the generalized entropy is

$$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$

and:

• $X$: codimension-2 bulk surface homologous to $A$,
• $\mathrm{Area}(X)$: geometric area of $X$,
• $S_{\mathrm{bulk}}[\Sigma_X]$: von Neumann entropy of the quantum fields in the region $\Sigma_X$ bounded by $X$ and $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$0.

Extremality requires

$$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$1

for all local deformations $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$2. This introduces a “quantum correction” to the usual minimal surface equation in gravity, often interpreted as an entropic force counterbalancing the geometric mean curvature $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$3 (Engelhardt et al., 2014, Mahajan, 4 Feb 2025).

2. Structure of the Prescription: Extremization, Minimization, and Maximin

Quantum extremal surfaces are determined by first finding all surfaces $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$4 anchored to $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$5 that solve the extremality equation, then selecting the one with the smallest $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$6. The existence and uniqueness of the minimal-QES can be established using a quantum generalization of the maximin construction. In this construction, for fixed boundary data, one minimizes $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$7 over all surfaces on a given Cauchy slice, then maximizes this minimum over all slices. Under the quantum focusing conjecture (QFC), this surface coincides with the minimal QES and ensures entanglement wedge nesting and strong subadditivity (Akers et al., 2019).

The same logic straightforwardly extends to hybrid entropy functionals incorporating nonholographic systems, leading to the inclusion of “quantum islands” in evaporation models.

3. Quantum Extremality Under Perturbations and Operator Formulation

Perturbative analysis leads to a “quantum extremal deviation equation”: $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$8 where $$S(A) = \min_{X \sim \partial A} \text{Ext}_X \left[ S_{\mathrm{gen}}[X] \right]$$9 is the classical Jacobi operator for normal deformations, and $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$0 denotes covariant functional derivatives of the von Neumann entropy. This framework allows stability analysis of QES and connects geometric constraints (focusing, wedge nesting) to properties of the nonlocal entropy kernel (Engelhardt et al., 2019).

At leading order, extremality is governed by the area term, subleading corrections produce FLM-type terms, and at higher orders, the extremal surface and the entropy kernel shift nontrivially—a genuinely quantum effect (Engelhardt et al., 2014).

4. Replica Trick, One-Shot Refinements, and Weighted Multi-Surface Formulas

The replica trick provides a path-integral derivation of QES via analytic continuation of $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$1 geometry. At finite $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$2, the naive saddle-point extremization becomes subtle as $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$3, requiring careful summation over configurations. Recent work derives an exact “multi-surface” formula,

$$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$4

with normalized weights $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$5, describing the entropy as a weighted average over all QES candidates (Khodahami et al., 17 Jun 2025). In the limit where one candidate dominates, this reduces to the standard prescription.

Furthermore, the sharpness of phase transitions predicted by QES is corrected by one-shot quantum information theoretic effects, governed by smooth min- and max-entropies $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$6, $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$7. The refined criterion for a phase transition is

• $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$8: the “no-island” QES dominates.
• $$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4 G_N} + S_{\mathrm{bulk}}[\Sigma_X]$$9: the “island” QES dominates.
• Intermediate regime: entropy transitions smoothly, and the formula must account for the detailed entanglement spectrum (Akers et al., 2020, Wang, 2021).

This formalism yields the correct reproduction of the Page curve (early/late-time entropy transition) in black hole evaporation and applies to both gravitational and non-gravitational systems coupled to gravity.

5. Defect Extremal Surfaces and Island-Generalized Entropy

In bulk geometries with co-dimension 1 defects (e.g., end-of-the-world branes in AdS/BCFT setups), the generalized entropy includes both a geometric area term and an additional entropic piece from the defect QFT on the brane: $X$0 The stationarity conditions $X$1, $X$2 enforce both bulk and defect quantum extremality (Li et al., 2021, Deng et al., 2020). For intervals in AdS$X$3/BCFT$X$4, this prescription exactly matches the standard QES/island formula after suitable decomposition (RS reduction, transparent boundary insertion), guaranteeing consistency between the bulk defect formulation and boundary entanglement island rules. The defect extremal surface method also generalizes to mixed-state reflected entropy via appropriate replica-index conventions.

6. Applications: JT Gravity, Cosmological Backgrounds, Modular Bootstrap

In JT gravity, the QES lies just outside the black hole horizon, with its stationarity condition corresponding to the microcanonical first law of a nested Rindler wedge (Pedraza et al., 2021). The generalized entropy incorporates Wald entropy, Polyakov term, and time-dependent von Neumann entropy, with the QES encoding the quantum corrections precisely (Mahajan, 4 Feb 2025).

In cosmological settings, the QES method accommodates effective 2D reductions and can be used to study the absence or structure of islands in singular backgrounds: in AdS-Kasner and dS/FRW, islands are generically absent or have singular on-shell entropy, while in null-Kasner there are formal extremal solutions with diverging $X$5 near the singularity (Goswami et al., 2021).

Recent developments also use CFT modular-flow techniques, extremality equations, and explicit Witten-diagram calculations to match CFT entanglement entropy at $X$6 to holographic QES results, including the canonical energy term and shape-deformation effects (Bhattacharya et al., 12 Dec 2025). There are explicit bootstrap correspondences between CFT OPE data and the QES expansion—proving, for example, that the area operator encodes the Virasoro vacuum block up to $X$7 (Belin et al., 2021).

7. Physical Consequences: Entanglement Wedges, Page Curve, Quantum Barriers

The QES defines the boundary of the so-called quantum entanglement wedge, which encodes the reconstructable bulk region given access to a boundary subregion. Quantum extremal surfaces are constrained to lie outside the causal wedge and may obey “quantum barrier” constraints, enforcing the generalized second law. The QES method resolves the black hole information paradox by predicting the emergence of “islands” that bound the entropy growth of Hawking radiation and by producing the appropriate Page curve (Engelhardt et al., 2014, Akers et al., 2019).

The entanglement wedge nesting property and strong subadditivity become rigorous theorems for QES in semiclassical gravity (Akers et al., 2019). Moreover, the defect extension to e.g., reflected entropy, exhibits the precise correspondence of quantum extremal surfaces to mixed-state quantum measures in brane-worlds and BCFT scenarios—offering compelling evidence for the universality and robustness of QES methods (Li et al., 2021).

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• (Engelhardt et al., 2014) Engelhardt, Wall: "Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime"
• (Akers et al., 2019) Akers, Engelhardt, Wall: "Quantum Maximin Surfaces"
• (Engelhardt et al., 2019) Engelhardt, Fischetti: "Surface Theory: the Classical, the Quantum, and the Holographic"
• (Deng et al., 2020) Chu, Liu: "Defect extremal surface as the holographic counterpart of Island formula"
• (Li et al., 2021) Li, Yuan, Zhou: "Defect Extremal Surface for Reflected Entropy"
• (Wang, 2021) Liu, Su: "The refined quantum extremal surface prescription from the asymptotic equipartition property"
• (Khodahami et al., 17 Jun 2025) Khodahami, Azizi: "A revision to the QES prescription"
• (Akers et al., 2020) Akers, Penington: "Leading order corrections to the quantum extremal surface prescription"
• (Bhattacharya et al., 12 Dec 2025) Bhattacharya, Parrikkar: "Modular Witten Diagrams and Quantum Extremality"
• (Belin et al., 2021) Belin, Colin-Ellerin: "Bootstrapping Quantum Extremal Surfaces I: The Area Operator"
• (Mahajan, 4 Feb 2025) Mahajan: "Lectures on Quantum Extremal Surfaces and the Page Curve"
• (Pedraza et al., 2021) Ouyang, Stoica: "Semi-classical thermodynamics of quantum extremal surfaces in Jackiw-Teitelboim gravity"
• (Goswami et al., 2021) Manu, Narayan, Paul: "Cosmologies, singularities and quantum extremal surfaces"
• (Wong, 2022) Wong: "A note on the bulk interpretation of the Quantum Extremal Surface formula"
• (Chen et al., 2020) Rozali, Sully, Van Raamsdonk et al.: "Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane"