The Quantum Extremality Condition

A quantum extremal surface is a stationary point of the renormalized generalized entropy, not of the area and bulk entropy separately. Its defining equation balances the geometric response to a local normal displacement against the state-dependent shape derivative of quantum entropy. This page derives that balance in a regulated semiclassical theory and shows how the classical extremal surface is displaced at first quantum order.

Required background. Quantum Information supplies density operators, von Neumann entropy, and the subsystem or algebra whose shape derivative is varied.

Helpful background. Semiclassical Gravity supplies renormalized generalized entropy and the loop expansion; Quantum Fields in Curved Spacetime supplies state-dependent entanglement on a prescribed geometry; Black-Hole Geometry and Mechanics supplies codimension-two geometry and normal deformations.

Local normal variations

Fix a state $\rho$, an anchored candidate surface $X$, and the associated regulated bulk algebra $\mathcal A(a_X)$. Let $y$ denote coordinates on $X$ and let $n_a^\mu$, $a=0,1$, span its normal plane. A deformation is

$$\delta X^\mu(y)=\zeta^a(y)n_a^\mu(y).$$

To avoid hiding a convention-dependent sign in a mean-curvature symbol, define the geometric response directly by

$$\mathcal K_a(y) := \frac{1}{\sqrt h} \frac{\delta\operatorname{Area}(X)}{\delta X^a(y)}.$$

Here $h$ is the determinant of the induced metric on $X$. The entropy response is similarly

$$\mathcal S_a(y) := \frac{1}{\sqrt h} \frac{\delta\bigl(S_{\mathrm{bulk,reg}}+S_{\mathrm{ct}}\bigr)} {\delta X^a(y)}.$$

Both derivatives are taken with the state preparation, anchoring data, regulator prescription, and asymptotic boundary region held fixed. The matter derivative is generally nonlocal: moving $X$ changes which operator algebra belongs to $a_X$.

Deriving quantum extremality

For Einstein gravity the first variation is

$$\delta S_{\mathrm{gen}} = \int_X d^{d-2}y\,\sqrt h\, \zeta^a(y) \left[ \frac{\mathcal K_a(y)}{4G_{\mathrm{ren}}} +\mathcal S_a(y) \right].$$

Because the allowed test functions $\zeta^a(y)$ are independent, stationarity for every local deformation gives the quantum extremality condition

$$\boxed{ \frac{\mathcal K_a}{4G_{\mathrm{ren}}}+\mathcal S_a=0 } \qquad (a=0,1).$$

Equivalently, one may define a generalized expansion in either null-normal direction $k^\mu$ by

$$\Theta_{\mathrm{gen}}^{(k)}(y) := \frac{4G_{\mathrm{ren}}}{\sqrt h} \frac{\delta S_{\mathrm{gen}}}{\delta\lambda_k(y)},$$

so a QES has vanishing generalized expansion along both independent null normals. Vanishing in only one null direction defines a quantum-marginal surface, which need not be a QES. This is the local definition given in Engelhardt and Wall 2015, §§2–3. The area and bulk-entropy terms arise together at first quantum order in Faulkner, Lewkowycz, and Maldacena 2013, §2.2.

The equation is finite only for the renormalized sum. A cutoff change can shift $\mathcal K_a/(4G_{\mathrm{ren}})$ and $\mathcal S_a$ individually while preserving their sum. Extremizing a bare area against an independently subtracted entropy would therefore be scheme dependent.

Perturbative displacement from a classical surface

Let $X_0$ be a nondegenerate classical extremal surface, so the first area variation vanishes there. Write its normal displacement as $\eta^a(y)$. Define $H_{ab}(y,y'):=\delta\mathcal K_a(y)/\delta X^b(y')$ at $X_0$, with the coordinate integration measure used below. This is the area Hessian with the output factor $1/\sqrt h$ included, so no hidden measure factor is needed. To first nontrivial order,

$$\mathcal K_a[X_0+\eta](y) = \int_X d^{d-2}y'\, H_{ab}(y,y')\eta^b(y')+O(\eta^2).$$

Evaluating the entropy response on $X_0$ then gives

$$\int_X d^{d-2}y'\, H_{ab}(y,y')\eta^b(y') =-4G_{\mathrm{ren}}\mathcal S_a[X_0](y),$$

and hence $\eta=O(G)$ when $H^{-1}$ exists and the matter response is $O(G^0)$. This makes the balance transparent: quantum entanglement supplies a force in surface space, while the classical area Hessian supplies the restoring response.

If $H$ has a zero mode, this expansion is not valid. A small entropy term can then produce an order-one displacement or split a degenerate family, and one must treat the collective coordinate or competing saddles nonperturbatively.

Checks, interpretation, and limits

Classical check. When the complete matter-plus-counterterm shape derivative vanishes, quantum extremality reduces to $\mathcal K_a=0$. A regulated product state at one cut is not enough to establish this: neighboring cuts generally have different entanglement, and an exact product across a continuum cut need not be a regular semiclassical state.

Symmetry check. A symmetry that reverses a normal deformation while preserving the state, anchoring, and entropy region forces the first variation in that direction to vanish. Both independent normal directions must be constrained to conclude quantum extremality. A time-reflection symmetry alone fixes the time-normal variation; it does not remove the need to extremize in the spatial normal direction.

Dimensional check. $S_{\mathrm{gen}}$ is dimensionless. The two terms in its functional derivative have the same dimension because $G$ converts an area variation into an entropy variation.

The equation only establishes stationarity. It does not enforce anchoring or homology, choose between disconnected topologies, prove stability, or select the least generalized entropy. It also assumes a semiclassical geometry and a well-defined regulated algebra. Generic QFT foundations and the renormalization of gravitational couplings remain in their specialist treatments rather than being rederived here.

In particular, a QES is an extremum and need not be the minimum that dominates the entropy prescription. A local maximum and an unstable saddle can solve exactly the same first-variation equation. One must compare the renormalized values of all admissible stationary branches, using a common state and subtraction scheme; the smallest branch is selected only after that separate global comparison. Near a branch merger, the Hessian test and the perturbative displacement formula can both fail.

Exercises

Near a classical extremum, suppose a single collective coordinate $x$ gives

$$S_{\mathrm{gen}}(x) =S_*+\frac{\kappa}{8G}(x-x_0)^2+s_1(x-x_0), \qquad \kappa\gt0.$$

Find the QES and its generalized entropy. What signals failure of the perturbative result?

Solution

Differentiation gives $\kappa(x-x_0)/(4G)+s_1=0$, so $x_{\mathrm{QES}}=x_0-4Gs_1/\kappa$. Substitution yields $S_{\mathrm{gen}}(x_{\mathrm{QES}})=S_*-2Gs_1^2/\kappa$. The shift is perturbatively small when $\lvert4Gs_1/\kappa\rvert$ is small compared with the geometric scale on which the expansion was made. A small or vanishing $\kappa$, or an entropy derivative scaling like $1/G$, invalidates that conclusion and requires a treatment of degeneracy or competing branches.

References

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