Quantum Extremal Surfaces

A quantum extremal surface (QES) is stationary under local deformations of the renormalized generalized entropy. One first extremizes generalized entropy on every admissible branch and then minimizes over those extrema to select the dominant entropy saddle. In the semiclassical large-N expansion, the QES is an order-$G_N$ correction to a nondegenerate classical extremal surface; near degeneracy, that perturbative description can fail.

Required background. Quantum Information supplies von Neumann entropy and algebraic subsystems, while Gauge Theory and String-Theory Preparation supplies large-N counting and the holographic region/surface map. Helpful background. Quantum Gravity supplies the semiclassical expansion, and Black-Hole Information supplies the general Page-curve and reconstruction questions that consume QES data.

Generalized entropy as a surface functional

Fix a boundary region $A$, a prepared semiclassical bulk state, an anchoring and homology class, and a common regulator. For an admissible codimension-two surface $X$ with associated bulk region $r_X$, define

$$S_{\rm gen}[X] =\frac{\operatorname{Area}(X)}{4G_{\rm ren}} +S_{\rm bulk,ren}(r_X) +S_{\rm local}[X].$$

The local term contains higher-curvature, edge, and counterterm contributions required by the chosen effective theory, with a fixed allocation between this term and bulk entropy. The area uses the semiclassical metric to the retained order, including quantum backreaction when required. Only the complete renormalized functional is meaningful: changing the cutoff shifts bulk entropy and geometric couplings separately. The state, boundary region, regulator, and asymptotic sources are held fixed while $X$ moves. Write the bulk dimension as $d+1$, so $X$ has dimension $d-1$.

An extremum is not yet the entropy. Denote by $\operatorname{Ext}(A)$ the real, admissible, homologous stationary branches. The semiclassical prescription is

$$S(A)=\min_{X\in\operatorname{Ext}(A)}S_{\rm gen}[X]$$

to the retained order. Thus extremization is a local variational condition, whereas minimization is a global comparison between branches. A local maximum can satisfy the QES equation and still be subdominant. Engelhardt and Wall formulate this quantum-corrected surface prescription and its causal role in Engelhardt and Wall 2015, §§ 2–4.

Deriving the quantum extremality equation

Let $n_a^\mu$, $a=0,1$, span the normal plane of $X$, and deform the embedding by $\delta X^\mu=\zeta^a n_a^\mu$. Define responses without fixing a sign convention for the mean-curvature vector:

$${\cal K}_a(y)=\frac{1}{\sqrt h} \frac{\delta\operatorname{Area}(X)}{\delta X^a(y)}, \qquad {\cal S}_a(y)=\frac{1}{\sqrt h} \frac{\delta[S_{\rm bulk,ren}+S_{\rm local}]}{\delta X^a(y)}.$$

The first variation is therefore

$$\delta S_{\rm gen} =\int_X d^{d-1}y\,\sqrt h\,\zeta^a(y) \left[\frac{{\cal K}_a(y)}{4G_{\rm ren}}+{\cal S}_a(y)\right].$$

Because the allowed test functions are independent in both normal directions, stationarity gives

$$\frac{{\cal K}_a}{4G_{\rm ren}}+{\cal S}_a=0, \qquad a=0,1.$$

The entropy response is a shape derivative of a quantum state’s algebra, generally nonlocal along $X$. Treating it as an ordinary local pressure is only a mnemonic. The area and bulk-entropy contributions at first quantum order arise together in the replica analysis of Faulkner, Lewkowycz, and Maldacena 2013, § 2.2.

Perturbative displacement from HRT

Let $X_0$ extremize area in the same metric used in $S_{\rm gen}$ and write the additional entropy-induced normal displacement as $\eta^a(y)$. If one instead expands about the uncorrected leading metric, its order-$G_N$ backreaction supplies a separate source in this displacement equation. Define $H_{ab}(y,y')$ as the linear-response kernel of $\mathcal K_a$ relative to the displayed coordinate measure $d^{d-1}y'$. After fixing tangential reparametrizations and anchoring conditions, suppose this normal Jacobi operator has an inverse on the allowed deformations. Then

$${\cal K}_a[X_0+\eta](y) =\int_{X_0}d^{d-1}y'\, H_{ab}(y,y')\eta^b(y')+O(\eta^2).$$

Evaluating the entropy response on $X_0$ yields

$$\int_{X_0}d^{d-1}y'\, H_{ab}(y,y')\eta^b(y') =-4G_{\rm ren}{\cal S}_a[X_0](y),$$

so $\eta=O(G_N)$ for an order-one entropy derivative. This recovers the classical surface when $G_N{\cal S}_a\to0$ and supplies an independent order-counting check. If the Hessian has a zero mode, a formally small quantum term can cause an order-one displacement, split a symmetric family, or exchange saddle dominance. Then the inverse-Hessian formula must not be used.

Branch selection, causal structure, and limits

Anchoring and homology are imposed before extremization. A surface on the wrong homology class cannot win merely because it has smaller generalized entropy. Each branch must also use the same renormalization scheme and bulk state; otherwise its finite values are not comparable. At a simple crossing, the selected entropy is continuous while a parameter derivative may jump. Near topology change, analytic continuation and saddle enumeration require special care.

Quantum extremality is compatible with semiclassical causal constraints under additional generalized-entropy assumptions, but the local equation alone does not prove them. Nor does it provide a microscopic Hilbert-space factorization, a nonperturbative definition of the gravitational path integral, or control through a high-curvature singularity. It is a saddle prescription within semiclassical gravity.

Three checks help prevent overinterpretation. If the bulk entropy shape derivative vanishes, the equation reduces to classical extremality. A reflection symmetry that reverses each normal can force both responses to vanish, producing a candidate but not proving dominance. Finally, varying a bare entropy while holding renormalized area couplings fixed produces cutoff-dependent motion and signals an inconsistent scheme.

Exercises

Compute a shifted QES and compare branches. Let one branch have $S_1(x)=100+(x-1)^2/(8G_N)+3(x-1)$ with $G_N=0.01$, and let an already-extremized competitor have $S_2=99.95$. Find the stationary point and decide which branch dominates.

Solution

Differentiation gives $(x-1)/(4G_N)+3=0$, hence $x_*=1-12G_N=0.88$. Substitution gives $S_1(x_*)=100-18G_N=99.82$. This is smaller than $S_2=99.95$, so the shifted first branch dominates. The calculation illustrates both stages: solving the derivative equation finds a QES, while comparing the stationary generalized entropies selects the entropy saddle. If the quadratic coefficient vanished, the computed shift would not be perturbatively controlled and this conclusion could not be trusted.

References

• Engelhardt, Netta, and Aron C. Wall. “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime.” Journal of High Energy Physics 01 (2015): 073. DOI. Open PDF.
• Faulkner, Thomas, Aitor Lewkowycz, and Juan Maldacena. “Quantum Corrections to Holographic Entanglement Entropy.” Journal of High Energy Physics 11 (2013): 074. DOI. Open PDF.