Generalized Entropy and Quantum Extremal Surfaces

Generalized entropy combines matter entropy with geometric entropy terms in a common renormalized prescription. It connects horizon thermodynamics to quantum extremal surfaces, while distinguishing stationarity, monotonicity, and entropy-reconstruction claims.

Enter this chapter

Begin with Black-Hole Thermodynamics at the QFT Interface to separate horizon temperature, flux, and entropy as physical quantities.

Helpful background. Replica Trick and Branched Geometries and Entropy Counterterms and Renormalization Ambiguities develop the definitions used by the chapter’s opening pages.

Topics

1. Black-Hole Thermodynamics at the QFT Interface — Separates Hawking temperature, state-dependent flux, and geometric entropy.
2. Fine-Grained, Coarse-Grained, and Algebraic Entropy — Fixes the entropy object before calculations begin.
3. Horizon Entanglement Entropy — Derives the local ultraviolet structure across a smooth horizon.
4. Relative Entropy and Modular Horizon Laws — Develops finite algebraic comparisons and known modular generators.
5. Generalized Entropy and UV Renormalization — Combines matter and gravitational counterterms.
6. Species, Gauge Edges, and Contact Terms — Makes the algebra and edge prescription explicit.
7. Noether-Charge Entropy and Higher-Curvature Terms — Derives stationary geometric entropy from the action.
8. Replica Constructions on Fixed and Semiclassical Backgrounds — States the boundary, saddle, symmetry, and continuation assumptions.
9. Conical Entropy and Effective-Action Variations — Tracks bulk, surface, and contact contributions.
10. Semiclassical First Laws and Physical-Process Variations — Distinguishes equilibrium variation from flux-driven change.
11. The Generalized Second Law: Hypotheses and Proven Scope — States precisely which causal-horizon results are proved.
12. Quantum Extremal Surfaces: Renormalized Semiclassical Definition — Defines renormalized stationarity and perturbative displacement.
13. Semiclassical Black-Hole Information: Scope and Limits — Identifies what evaporation arguments require beyond this chapter.

How the ideas fit together

The ultraviolet dependence of matter entanglement must be combined with the renormalization of gravitational surface terms. Once that generalized entropy is defined, varying the surface gives the quantum-extremality condition Engelhardt and Wall 2015, §§ 2–3.

Stationarity identifies candidate surfaces; it does not by itself select a dominant saddle or prove a boundary entropy formula. The generalized second law instead concerns monotonicity under stated horizon and state assumptions. Keeping these questions separate prevents a local entropy variation from being mistaken for a complete black-hole information calculation.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Object or statement	State and algebra	Regulator and geometry	Licensed conclusion	Status	Failure or further method needed
Hawking thermodynamics	Declared black-hole state; exterior observables	Stationary nonextremal Killing horizon; renormalized stress tensor	$T_H=\kappa/(2\pi)$, flux after greybody scattering, stationary first-law data	QFT calculation plus classical geometry	A rapidly evolving trapping horizon lacks the required symmetry
Fine/coarse entropy	State on an algebra; coarse constraints specified	UV regulator only for subregion entropy	Distinguish von Neumann, relative, and thermodynamic entropy	Definition/algebraic theorem	Type-III local algebras defeat naive tensor-factor entropy arithmetic
Horizon entanglement	Hadamard-like short-distance state; chosen algebra	Smooth cut; explicit UV regulator	Local area and curvature divergences; finite differences when controlled	QFT calculation	Regulator-dependent constants are not observables
Relative/modular law	Normal states on one algebra; modular domains valid	Often regulator free	Positivity, monotonicity, $\delta S=\delta\langle K\rangle$ near a reference state	Operator-algebra theorem	A geometric $K$ is unavailable for a generic region/state
Renormalized $S_{\rm gen}$	Matter state and algebra fixed	Matter and gravitational counterterms matched	Finite scheme-consistent generalized-entropy differences/variations	Semiclassical EFT construction	Shifting only one sector produces spurious scheme dependence
Gauge/edge entropy	Center or edge prescription declared	Gauge-compatible regulator and contact terms	Comparable entropy within that prescription	Algebra-dependent construction	Changing the algebra while holding counterterms fixed changes the question
Noether-charge entropy	Classical stationary solution	Diffeomorphism-invariant renormalized action; bifurcate Killing surface	Wald entropy and stationary first law	Classical theorem/semiclassical coupling input	Dynamical cuts carry ambiguity and extra flux terms
Replica entropy	State preparation and replica boundary data	Integer covers; action, defects, and saddles specified	$S=(n\partial_n-1)I_n\rvert_{n=1}$ if continuation exists	Path-integral calculation	Integer data need not select the derivative at $n=1$
Conical effective action	Matter boundary conditions and zero modes fixed	Smoothed cone or distributional prescription	Bulk/surface/contact contribution to entropy	One-loop calculation	Cone and UV limits may not commute
First law	Perturbation of a stationary reference state	Linear order; suitable horizon and asymptotics	Equality among charge, flux, area, and modular variations	Perturbative theorem	Shear, expansion, and quadratic entropy production matter beyond linear order
Generalized second law	Horizon algebra and regular state satisfy stated axioms	Causal horizon; joint renormalization; semiclassical regime	Monotonic $S_{\rm gen}$ for the theorem’s slices	Theorem in specified field classes	Apparent horizons and generic interacting theories are not automatic corollaries
Quantum extremal surface	$\rho$ and $\mathcal A_{\rm out}(X)$ declared	Admissible anchored/homologous surface class; renormalized functional	$\delta_XS_{\rm gen}=0$ and controlled displacement	Semiclassical definition/calculation	Stationarity alone neither minimizes nor supplies an island formula
Information claim	Exterior/radiation algebra and operational task fixed	Evaporation approximation and gravity input stated	A scoped entropy or recovery statement	Model-dependent/open	Mean Hawking flux alone cannot establish a Page curve or microscopic unitarity

Where to go next

Continue to Energy Conditions, Inequalities, and Focusing to identify which classical energy condition a quantum expectation value violates.

References

• Engelhardt, N., and A. C. Wall, “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,” Journal of High Energy Physics 2015, 073 (2015), doi:10.1007/JHEP01(2015)073.