How quantum corrections move an extremal surface

A surface divides the bulk quantum fields too

Chapter 4 assigned a geometric length to a curve. Put quantum fields in the bulk and the two sides of the curve can be entangled. A pair of entangled excitations on opposite sides adds uncertainty to either side alone, just as the two qubits did in Chapter 2. A correction to boundary entropy must respond to this bulk information.

Treat gravity as a static semiclassical background and include one-loop matter fields with their backreaction. With $\hbar=1$ and fixed classical length scales, area contributes at order $1/G_N$ and the one-loop field correction at order $G_N^0$ . In their static setting, Faulkner, Lewkowycz and Maldacena express the next contribution using bulk entanglement and additional local terms. The area term includes backreaction at this order. The additional local terms include Wald-like corrections and renormalization contributions that cancel short-distance divergences.

S_{\rm gen}[X]=\frac{\langle A[X]\rangle}{4G_N}+S_{\rm bulk}[\Sigma_X]+S_{\rm local}[X]

The surface

X

and boundary region enclose

\Sigma_X

. Area is an expectation value in the chosen state, including backreaction at the retained order. All terms use compatible renormalization prescriptions; entropy is in nats.

Source: Thomas Faulkner · Netta Engelhardt and Aron C. Wall

The old surface need not remain stationary

Moving the surface changes both its area and the quantum region it divides. At the classical extremum, the first area variation vanishes. The first bulk-entropy variation generally does not. A new stationary surface appears nearby: the quantum extremal surface. Surfaces must be boundary-anchored and homologous to the boundary region. Extremize in all normal directions before comparing candidates. A least-entropy saddle dominates when the entropy separation is much larger than one nat; near degeneracy, several saddles can contribute comparably.

For a local worked model in AdS3, use the curvature radius $L$ to define dimensionless $g=G_N/L$ and $a_0=A_0/L$ . Let $q$ be a dimensionless displacement mode. Write the area contribution as $a_0/(4g)+Kq^2/(2g)$ and the remaining entropy as $S_0+bq$ . Choose $K>0$ for a classical local minimum and parameters giving sensible total entropy in the neighborhood. The non-area entropy slope $b$ shifts the stationary point to $q_*=-gb/K$ .

Worked example: in one consistent dimensionless normalization, set $g=0.01$ , $K=2$ and $b=3$ . Then $q_*=-0.015$ . The entropy at the shifted point is lower by $0.0225$ nats. Displacement is first order in the small coupling; reevaluating entropy at that position changes it one order beyond the one-loop term. This explains why evaluating bulk entropy on the classical surface suffices at FLM accuracy.

\frac{\delta S_{\rm gen}}{\delta X}=0,\qquad q_*=-\frac{gb}{K},\qquad S_{\rm gen}(q_*)-S_{\rm gen}(0)=-\frac{gb^2}{2K}

These displacement formulas belong to the authored local quadratic example and require a small displacement. They explain perturbative orders without solving a particular black-hole geometry.

Source: Netta Engelhardt and Aron C. Wall

Two candidates can produce a Page-shaped curve

Consider a black hole exchanging information with an external radiation system. The early semiclassical Hawking-pair calculation gives increasing radiation entropy. If complete evaporation is unitary from a pure initial state, the final radiation should also be pure. A quantum extremal surface calculation can admit a different radiation candidate whose bulk region includes an island and pays a corresponding area contribution. Competition changes the dominant answer.

First isolate the mathematics of competition. Let $u\in[0,1]$ denote schematic progress. Assign candidate entropies $s_1(u)=20u$ and $s_2(u)=12(1-u)$ nats. They meet at $32u=12$ , giving $u_*=3/8$ and entropy $7.5$ . Their minimum rises and then falls. These functions are specified by hand to isolate the competition between candidates.

Each branch can be smooth while the selected answer develops a kink. A physical derivation must first establish that both candidates are admissible quantum extremal saddles and specify the coupling between radiation and the gravitational region. Near the crossing, multiple-saddle contributions at finite coupling can also correct the sharp kink.

s_{\rm schematic}(u)=\min\{20u,12(1-u)\},\qquad u_*=\frac38

The lower branch of two assigned entropy functions illustrates the change of dominant candidate.

Authored two-candidate entropy illustration. Cyan increases as twenty times progress; gold decreases as twelve times remaining progress. The white curve selects the smaller value. These functions are specified for the example.

Source: Geoffrey Penington

Restore the physical assumptions

Penington’s evaporating setup uses absorbing boundary conditions so energy can leave an AdS black hole. It differs from the fixed pure-AdS signal geometry in Chapter 1. The calculation involves quantum extremal surfaces, a boundary radiation system and a specified semiclassical state. Reconstruction can depend on the code subspace and the requested precision.

This theoretical development connects radiation entropy, bulk regions and recoverable information in calculable settings. Microscopic decoding complexity, dynamics beyond controlled approximations and extensions to the spacetime of our universe require further methods. A small tensor or fixed-background browser model helps with particular steps. The final chapter identifies exactly which steps our calculations reproduce.

Source: Geoffrey Penington · Xi Dong

Try the reasoning

Use classical candidate competition to pose the quantum question

Bring two intervals together and locate the crossing of their two candidate total lengths.
Explain why the selected pairing switches although each individual candidate varies continuously.
Add a constant bulk-entropy contribution to one candidate on paper. Predict the direction of the shifted crossing. The experiment itself does not calculate bulk quantum fields.

Exercises and explanations

In the displacement example, what happens for $b=0$ ? What happens if $g$ is halved with other parameters fixed?
Read the explanation
Zero slope gives $q_*=0$ , so the old surface remains stationary. Halving $g$ halves both the displacement and the entropy difference caused by relocation, consistent with their perturbative order.
Raise the second schematic entropy candidate by two nats. Where is the new crossing, and what has this calculation established?
Read the explanation
Solve $20u=14-12u$ to find $u=7/16$ and entropy $8.75$ nats. This tests how specified candidate functions change their crossing. Their physical existence still needs a separate derivation.

Sources and recalculated scope

The course derives its worked examples and draws its own schematics. The numerical references recalculate specified analytic geometry and a finite five-qubit encoding. The source locations below distinguish those calculations from theoretical reading topics.

Xi Dong, Daniel Harlow and Aron C. Wall, Bulk Reconstruction in the Entanglement Wedge in AdS/CFT, arXiv:1601.05416v2; PRL 117, 021601 (2016)
Sections I–II and Figure 1 define causal and entanglement wedges and a perturbative code subspace. Sections III–IV state the relative-entropy assumptions behind reconstruction. We explain these assumptions; the finite code checks are a separate exact model.
Thomas Faulkner, Aitor Lewkowycz and Juan Maldacena, Quantum corrections to holographic entanglement entropy, arXiv:1307.2892v2; JHEP 11 (2013) 074
Equations (1.1)–(1.3), section 2.2 and equation (2.15): bulk entanglement, area backreaction, Wald-like and counterterm contributions at one loop. This course explains the order of the expansion and keeps the renormalization terms; it does not evaluate a gravitational path integral.
Netta Engelhardt and Aron C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, arXiv:1408.3203v3; JHEP 01 (2015) 073
Equations (1.1), (3.1)–(3.2), section 3: generalized entropy, normal-direction extremization, the least-entropy prescription with its near-degeneracy qualification, and the relation to FLM. The quadratic displacement example is authored here and is not a solution of semiclassical Einstein equations.
Geoffrey Penington, Entanglement wedge reconstruction and the information paradox, arXiv:1905.08255v2; JHEP 09 (2020) 002
Sections 2.2–2.3 and 3: evaporating AdS black holes with absorbing boundary conditions, the Page transition and code-subspace/state-dependence conditions. Our two-candidate entropy plot is an explicitly schematic comparison, with no Hawking spectrum or evaporating spacetime simulation.