Quantum Extremal Surface Prescription

• Quantum extremal surfaces are computed by extremizing a generalized entropy functional that combines the classical area with quantum entanglement contributions.
• This prescription is pivotal in modeling black hole evaporation, reproducing the Page curve, and facilitating entanglement wedge reconstruction in holography.
• Recent refinements, including weighted summation over candidate surfaces, reinforce holographic entropy inequalities and support the quantum focusing conjecture.

The quantum extremal surface (QES) prescription is a fundamental tool in semiclassical gravity and holography, generalizing the Ryu–Takayanagi (RT) and Hubeny–Rangamani–Takayanagi (HRT) formulas by incorporating quantum corrections into the calculation of entanglement entropy for boundary regions in a holographic duality. Rather than extremizing the area functional alone, the QES prescription requires extremizing the generalized entropy, which includes both geometric and bulk quantum contributions. This concept is central to correctly capturing black hole entropy evolution, reproducing the Page curve in evaporating scenarios, and providing a precise operational definition for entanglement wedge reconstruction, holographic entropy inequalities, and quantum error correction.

1. Definition and Generalized Entropy Functional

The QES prescription defines the entanglement entropy of a boundary region $R$ via: $$S(R) = \underset{X \sim R}{\min} \, \underset{X}{\mathrm{ext}}~ S_{\mathrm{gen}}(X)$$ where $X$ is a codimension-2 bulk surface homologous to $R$. The generalized entropy $S_{\mathrm{gen}}$ is

$$S_{\mathrm{gen}}(X) = \frac{\langle A(X) \rangle}{4G\hbar} + S_{\mathrm{ent}}(\Sigma_X) + (\text{counterterms}),$$

with $A(X)$ the area operator, $S_{\mathrm{ent}}$ the von Neumann entropy of bulk quantum fields across $X$, and counterterms handling renormalization or higher-curvature contributions (Engelhardt et al., 2014).

The QES is any $X$ satisfying the quantum extremality condition: $\frac{\delta S_{\mathrm{gen}}}{\delta X^a} = 0,$ directly generalizing the extremality of the area functional in classical RT/HRT (Engelhardt et al., 2014, Dong et al., 2017). At leading order in the semiclassical expansion, this condition reproduces the FLM (Faulkner–Lewkowycz–Maldacena) formula, but it extends beyond to incorporate all orders in Planck’s constant and higher-curvature corrections.

2. Derivation, Variational Principles, and Modular Extremality

A rigorous derivation of the QES prescription employs the replica trick in the Euclidean gravitational path integral. Variation of the gravitational action under replica index and metric deformations yields the quantum extremality condition for the entangling surface (Dong et al., 2017): $$\delta_{\mathrm{diff}}\left( \frac{\langle A_{\mathrm{gen}} \rangle}{4G_N} + S_{\mathrm{bulk}} \right) = 0.$$ This variational framework extends to all orders in $G_N$ and applies to mixtures of states, giving rise to the modular extremal surface prescription for computing boundary modular Hamiltonians and relative entropy: $$K^{\text{bdy}}_\sigma = \frac{A^{X_\sigma}}{4G_N} + K^{X_\sigma}_{\text{bulk},\sigma},$$ with $$\delta_{X_\sigma}\langle K^{\text{bdy}}_\sigma\rangle = 0$$ at the modular extremal surface $X_\sigma$ (Dong et al., 2017).

3. Geometric Properties, Causality, and Barriers

Quantum extremal surfaces differ significantly from classical extremal surfaces in their placement and causal properties:

• Outward Placement: QESs are strictly outside the causal wedge $W_R$ of the boundary region $R$, lying spacelike to its causal surface, as required by the generalized second law (GSL). This ensures the generalized entropy at the QES is invariant under local, unitary operations in $R$—bulk regions behind the QES cannot be affected by such operations (Engelhardt et al., 2014).
• Barrier Surfaces: Bulk “barriers” are null splitting surfaces $M$ with every spacelike slice $Q$ quantum trapped—i.e., $(\delta S_{\mathrm{gen}}/\delta Q^a k^a) < 0$ for all $q\in Q$. No QES propagating from the exterior can cross or touch $M$. These barriers restrict the reconstructable region of the bulk and highlight “blind spots” inaccessible to extremal surface probes (Engelhardt et al., 2014).

4. Quantum Maximin Surfaces and Islands

A quantum maximin surface is constructed by minimizing $S_{\mathrm{gen}}$ on each Cauchy slice and then maximizing over slices, with the resultant surface equivalent to the minimal QES (i.e., the one realizing the global minimum of $S_{\mathrm{gen}}$ as required by the Engelhardt–Wall prescription) (Akers et al., 2019). This construction is robust even in scenarios where gradients of the bulk entropy are comparable to area variations and underpins:

• Entanglement Wedge Nesting: The entanglement wedge for a smaller region is contained within that of any larger region.
• Strong Subadditivity: Bulk subadditivity, together with the quantum maximin prescription, ensures holographic strong subadditivity for $S_{\mathrm{gen}}$.
• Quantum Extremal Islands: For gravitating systems coupled to nonholographic quantum systems, QESs (or islands) may become disconnected from the boundary but nonetheless enter into the computation of boundary entanglement entropy, explaining unitarity in black hole evaporation and reproducing the Page curve (Akers et al., 2019, Chen et al., 2020).

5. Stability, Elliptic Operators, and the Quantum Focusing Conjecture

Small perturbations of extremal surfaces are governed by generalizations of the Jacobi operator to quantum surfaces, with nonlocal contributions from the second functional derivative of the entropy: $$J(\eta_\perp)^a + 4G_N\hbar \int_\Sigma P^{ab}(p) \frac{\delta^2 S_\text{out}}{\delta \Sigma^c(p') \delta \Sigma^b(p)} \eta^c(p')\, dp' = \text{source terms}$$ (Engelhardt et al., 2019). The associated stability and perturbation theory reveal:

• Strong and Weak Stability: Defined by the positivity properties of the elliptic operator governing normal deformations of the QES.
• Quantum Focusing Conjecture (QFC): The second functional derivative of $S_{\mathrm{gen}}$ along any future-directed null generator is nonpositive, encapsulating the semiclassical consistency condition necessary for the QES prescription (Engelhardt et al., 2019, Bousso et al., 23 Oct 2024). Discrete max-focusing further refines the QFC by formulating nonexpansion with one-shot max-entropy measures, accommodating non-smooth geometries and ensuring robust wedge nesting and strong subadditivity (Bousso et al., 23 Oct 2024).

6. Impact on the Black Hole Information Problem and the Page Curve

In models of evaporating black holes (notably JT gravity coupled to CFT), QESs and the formation of islands provide a dynamical, self-consistent resolution to the information paradox. Before the Page time, the standard (no-island) QES dominates, yielding ever-growing entropy for the radiation, in accordance with Hawking’s prediction. After the Page time, a nontrivial QES appears (“island”), causing the radiation entropy to saturate at approximately twice the Bekenstein–Hawking entropy; this matches the unitary Page curve (Mahajan, 4 Feb 2025, Akers et al., 2019). The QES prescription is also manifest in explicit computations involving the extremization of the generalized entropy functional over possible island configurations, with transitions between multiple QES candidates leading to nontrivial entropy-phase transitions.

7. Recent Refinements and Weighted Summation over Multiple Surfaces

The original saddle-point-based derivation of the QES prescription via the replica trick assumes that the minimal surface dominates the path integral. Advances clarify that as $n \rightarrow 1$, the weighting of candidate surfaces by $\exp(-S(\Sigma_X))$ can lead to a breakdown of this approximation, making it necessary to compute a weighted sum over all possible surfaces (Khodahami et al., 17 Jun 2025): $$S = \frac{\sum_X \bigl[ S_{\mathrm{gen}}(X) + S(\Sigma_X) \bigr] e^{-S(\Sigma_X)}}{\sum_X e^{-S(\Sigma_X)}} + \ln\left( \sum_X e^{-S(\Sigma_X)} \right),$$ where $S(\Sigma_X)$ is the on-shell semiclassical action outside $X$ and the sum is over all possible X. In the limit where the distribution is sharply peaked, this reduces to evaluation at the dominant QES; otherwise, subdominant surfaces contribute nontrivially. This refinement ensures correct entropy evolution (e.g., the Page curve), accommodates transitions, and aligns with quantum information-theoretic approaches that use one-shot entropy measures and state-dependent wedge reconstruction (Wang, 2021, Akers et al., 2020).

8. Constraints, Holographic Inequalities, and Future Directions

The QES prescription, together with constraints on the bulk quantum state (such as the holographic entropy cone inequalities and the monogamy of mutual information), ensures that boundary entanglement properties satisfy the same structural inequalities as in the geometric RT limit (Akers et al., 2021). There is ongoing work to delineate the precise conditions on bulk entanglement and geometry necessary to interpolate between classical and fully quantum holographic entropy cones. Developments such as the discrete max-focusing conjecture streamline the axiomatic foundation, suggesting asymmetric, outward-directed, single-entropy-based approaches are sufficient for bulk reconstruction and wedge definition (Bousso et al., 23 Oct 2024).

Table: Key Mathematical Objects and Their Roles

Symbol or Formula	Meaning	Context/Role
$S_{\mathrm{gen}}(X)$	Generalized entropy	Sum of area and bulk entropy
$\frac{\delta S_{\mathrm{gen}}}{\delta X^a} = 0$	Quantum extremality condition	QES location equation
$$K^{\text{bdy}}_\sigma = \frac{A^{X_\sigma}}{4G_N} + K^{X_\sigma}_{\text{bulk},\sigma}$$	Modular extremal surface	Bulk dual of modular Hamiltonian
$$S = min_X [ Area(X)/(4G) + S_{\mathrm{ent}}(\Sigma_X) ]$$	Classical QES prescription	Boundary entropy via QES
$S = \sum_X P(X) S_{\mathrm{gen}}(X) + ...$	Weighted sum over surfaces	Refined prescription (Khodahami et al., 17 Jun 2025)
$S(\mathrm{radiation}) \sim \begin{cases}$growing$& t < t_{\mathrm{Page}} \$saturating$& t > t_{\mathrm{Page}} \end{cases}$	Page curve	Information paradox resolution

Significance: These quantities, together with conditions enforcing outward nonexpansion (QFC), strong subadditivity, and wedge nesting, collectively define the operational and structural backbone of the quantum extremal surface prescription in semiclassical gravity and holography.

• “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime” (Engelhardt et al., 2014)
• “Entropy, Extremality, Euclidean Variations, and the Equations of Motion” (Dong et al., 2017)
• “Surface Theory: the Classical, the Quantum, and the Holographic” (Engelhardt et al., 2019)
• “Quantum Maximin Surfaces” (Akers et al., 2019)
• “A revision to the QES prescription” (Khodahami et al., 17 Jun 2025)
• “Lectures on Quantum Extremal Surfaces and the Page Curve” (Mahajan, 4 Feb 2025)
• “Discrete Max-Focusing” (Bousso et al., 23 Oct 2024)
• “Quantum Extremal Surfaces and the Holographic Entropy Cone” (Akers et al., 2021)
• “The refined quantum extremal surface prescription from the asymptotic equipartition property” (Wang, 2021)
• “Leading order corrections to the quantum extremal surface prescription” (Akers et al., 2020)