Stretched Quantum Extremal Surfaces

• The paper shows that stretched quantum extremal surfaces are controlled refinements of QES, achieved via perturbative displacement equations and quantum extremal deviation analyses.
• Methodologies include operational string smearing, algebraic thickening, and tensor network discretization to regulate the surface near horizons or singularities.
• Stretched QES illuminate the interplay of generalized entropy, modular flow, and semiclassical backreaction, bridging ideal QES with practical regulatory frameworks.

Stretched quantum extremal surfaces are not a single uniformly defined object across the literature. The common substrate is the quantum extremal surface (QES): a codimension-2 surface that extremizes the generalized entropy, $S_{\mathrm{gen}}=\frac{A}{4G_N}+S_{\mathrm{bulk}}$ in semiclassical gravity, or the corresponding dilaton-plus-entropy functional in Jackiw–Teitelboim (JT) gravity. What is “stretched” differs by context. In some papers, stretching means a small perturbative displacement away from an exact QES; in others it means an operationally smeared entry boundary produced by finite string length, a shell of algebraic thickness where split property fails, a discretized surface pinned one row behind a tensor-network horizon, or a regulator-supported surface near a cosmological or black-hole horizon. This suggests that the expression is best treated as a family of closely related refinements of the sharp QES notion rather than a single canonical definition (Engelhardt et al., 2014, Engelhardt et al., 2019, Herderschee et al., 2 Oct 2025).

1. Foundational definition and the main senses of “stretching”

The foundational prescription is due to Engelhardt and Wall: for a boundary region $R$, the holographic entropy is computed by minimizing the generalized entropy over bulk codimension-2 surfaces $\Sigma$ homologous to $R$,

$$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$

with

$$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$

Quantum extremality is the vanishing of the first variation of $S_{\mathrm{gen}}$ under local deformations along both null normals; in this sense a QES is quantum marginally trapped in both directions (Engelhardt et al., 2014).

Within that framework, later literature uses “stretched” in several distinct but technically precise ways.

Sense of stretching	Representative formulation	Key diagnostic
Perturbative displacement	Small normal deformation of a QES driven by boundary, metric, or matter perturbations	Quantum extremal deviation equation
Operational stringy smearing	Probe-dependent delayed entry into an entanglement wedge	Crossover time, modular energy, fidelity
Algebraic thickening	Shell where split property fails at string scale	Stretched horizon $*$-algebra $A_{\mathrm{sh}}$
Discrete interior regulation	Surface pinned to a tensor-network row behind the horizon	Row-wise $S_{\mathrm{gen}}$ minimization
Regulated horizon/cosmology surface	Cutoff-supported or horizon-proximate extremal surface	Lag relation, near-horizon turning point

The perturbative meaning is explicit in “Surface Theory: the Classical, the Quantum, and the Holographic,” which does not itself use the term “stretched QES” but develops the equation of quantum extremal deviation governing small displacements of a QES under anchor changes, metric perturbations, and matter-state perturbations (Engelhardt et al., 2019). A second, more operational meaning appears in “Scattering strings off quantum extremal surfaces,” where a “stringy QES” is the effective target seen by a probe when finite $R$0, finite smearing, and sub-maximal chaos smooth the otherwise sharp null-separation criterion for entry into the entanglement wedge (Chandrasekaran et al., 2021). A third, algebraic meaning appears in “Stringy algebras, stretched horizons, and quantum-connected wormholes,” where stretched QES are defined as algebraically thickened boundaries supported on the shell where the split property fails at distances of order $R$1 (Herderschee et al., 2 Oct 2025).

A crucial qualification is that not every stretched object can serve as the exact entanglement wedge boundary. “Extremality as a Consistency Condition on Subregion Duality” argues in JT gravity coupled to a CFT that any candidate boundary with nonzero outward variation of $R$2 is inconsistent with complementary causal wedge exclusion under sufficiently strong complementary modular flow. In that sense, exact wedge boundaries must remain exactly quantum extremal, even if stretched constructions are useful approximations or regulators (Soni, 2024).

2. Perturbative geometry: quantum extremal deviation and microscopic displacement

The most systematic geometric account of stretched QES is the perturbative one. For a non-null surface $R$3, the classical deformation problem is controlled by the Jacobi or Simons stability operator $R$4, acting on normal deformations $R$5. In codimension two and a null basis $R$6, the classical deviation equations become a coupled elliptic system. Quantum extremality replaces extremality of area by extremality of generalized entropy,

$R$7

and the corresponding linearized displacement is governed by the equation of quantum extremal deviation (Engelhardt et al., 2019): $R$8

This packages stretched-QES kinematics into a sourced linear problem,

$R$9

where the quantum stability operator $\Sigma$0 consists of the classical elliptic operator $\Sigma$1 plus a multilocal entropy kernel. In the weakly stable regime, where zero is absent from the Dirichlet spectrum of $\Sigma$2, the Fredholm alternative yields existence and uniqueness of the linearized displaced surface for arbitrary boundary data and sources. In that sense, “stretching” is a controlled small-deformation problem rather than a distinct nonperturbative object (Engelhardt et al., 2019).

The near-horizon AdS Rindler example makes this concrete. For the bifurcation surface $\Sigma$3, the QES displacement $\Sigma$4 is written in terms of a Dirichlet Green’s function $\Sigma$5 on $\Sigma$6: $\Sigma$7

$\Sigma$8

These are explicitly described as the displacement of the HRT surface away from the horizon due to sources, and they provide a canonical perturbative model of a stretched QES (Engelhardt et al., 2019).

A complementary microscopic perspective appears in “Bootstrapping Quantum Extremal Surfaces I: The Area Operator.” In AdS$\Sigma$9/CFT$R$0, the QES displacement is expanded as $R$1, and in AdS-Rindler coordinates the shift is parameterized by

$R$2

The geometric part is controlled by the Virasoro identity block, while the non-identity blocks encode the bulk matter entropy contribution. To order $R$3, the paper shows that the identity block contributes solely to the area operator. This supplies a microscopic CFT meaning to the perturbative “stretching” of a classical RT surface into a QES (Belin et al., 2021).

3. Stringy, modular, and algebraic thickening

A distinct use of stretching is operational rather than variational. In “Scattering strings off quantum extremal surfaces,” a particle or string is inserted in a Rindler-like wedge, and one asks when boundary diagnostics detect that it has entered the entanglement wedge of a complementary region. In maximally chaotic theories, $R$4, the transition is sharp and occurs exactly at null separation from the QES: $R$5 For $R$6, finite string length, reduced Lyapunov exponent $R$7, and finite smearing $R$8 replace this by a delayed and smoothed crossover,

$R$9

The paper emphasizes that this “stretching” may be at least partly a property of the probe, via longitudinal string spreading, rather than a universal modification of the QES prescription itself (Chandrasekaran et al., 2021).

Modular transport yields yet another operational refinement. “Quantum Extremal Modular Curvature: Modular Transport with Islands” defines the Quantum Extremal Modular Curvature (QEMC) for JT gravity with islands. The exact boundary transport generator coherently drags both bath and island endpoints according to the QES condition,

$$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$0

and the corresponding modular curvature is generally non-local. In an OPE or effective single-interval limit, however, the QEMC becomes local and probes the bulk Riemann curvature near the island. The regulated interval on which this local curvature is sampled functions as an operational stretched-QES representative of the true QES (Aalsma et al., 2024).

The algebraic formulation is more radical. In the $$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$1, finite-string-tension regime, “Stringy algebras, stretched horizons, and quantum-connected wormholes” models the infinite tower of string modes as free fields with Hagedorn growth,

$$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$2

This growth violates modular nuclearity bounds and causes split property to fail once separations are of order $$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$3. The paper then defines stretched QES as the shell of thickness $$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$4 around a classical QES where factorization fails, together with a stretched horizon $$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$5-algebra $$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$6 supported on that shell. In this setting, the “surface” is no longer sharp in a geometric sense; it becomes an algebraically thickened boundary. Reflected entropy diverges when split fails, and the associated purified algebras can become type III$$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$7, which the paper interprets as a signature of quantum-connected wormholes between geometrically disjoint regions (Herderschee et al., 2 Oct 2025).

4. JT gravity, islands, tensor networks, and interior dynamics

Semi-classical JT gravity provides the cleanest explicit horizon-proximate stretched-QES solution. “Semi-classical thermodynamics of quantum extremal surfaces in Jackiw–Teitelboim gravity” shows that the semi-classical Wald entropy,

$$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$8

fully reproduces the generalized entropy, including the time-dependent von Neumann term, and that its extremization gives a QES lying just outside the classical horizon. In the Hartle–Hawking state the displacement is small,

$$S(R)=\min_{\Sigma\sim R}\,\mathrm{ext}\;S_{\mathrm{gen}}[\Sigma],$$9

so the QES is literally stretched outside the horizon. The same surface is the bifurcation surface of a nested AdS-Rindler wedge with temperature $$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$0, and the paper derives a semi-classical Smarr relation and first law for that nested wedge, identifying the generalized entropy as its thermodynamic entropy (Pedraza et al., 2021).

Island dynamics can make stretching transient and non-monotonic. “Ephemeral Islands, Plunging Quantum Extremal Surfaces and BCFT channels” studies finite intervals in Minkowski baths coupled to an eternal JT black hole. There the QES can start inside the horizon, emerge outside, and then plunge back in. In the high-temperature near-horizon regime, the QES positions satisfy

$$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$1

and the exit time for semi-infinite intervals is

$$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$2

The resulting “ephemeral islands” dominate only at intermediate times and produce a characteristic entropy dip that matches special disconnected OPE channels in a free-fermion BCFT description (Hollowood et al., 2021).

Tensor-network models supply a discretized interior counterpart. “Tensor networks for black hole interiors: non-isometries, quantum extremal surfaces, and wormholes” places effective interior qudits on rows of a hyperbolic tessellation. Candidate surfaces $$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$3 are pinned to row boundaries, with discrete generalized entropy

$$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$4

and local extremality determined by row differences such as

$$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$5

A candidate QES at row $$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$6 appears when $$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$7, and with $$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$8 this coincides with the non-isometry criterion for that row. The resulting QES are “stretched” in a discrete sense: they sit one row behind the horizon regulator $$S_{\mathrm{gen}}[\Sigma]=\frac{\langle A(\Sigma)\rangle}{4G_N\hbar}+S_{\mathrm{bulk}}[\Sigma]+S_{\mathrm{ct}}[\Sigma].$$9, move inward by jumps as rows are shed during evaporation, and are influenced by wormhole-like contractions to the radiation (Bueller et al., 2024).

5. Cosmological and de Sitter realizations

In cosmological singularity backgrounds, stretching typically means repulsion from a Big-Crunch region rather than approach to it. “Cosmological singularities, entanglement and quantum extremal surfaces” studies isotropic AdS Kasner and related 2D reductions. At the classical level, HRT surfaces dip radially but bend away from the singularity: at the turning point,

$S_{\mathrm{gen}}$0

so $S_{\mathrm{gen}}$1 is a local maximum and $S_{\mathrm{gen}}$2 for a surface anchored at boundary time $S_{\mathrm{gen}}$3. In the reduced 2D theory, the QES equations similarly drive both $S_{\mathrm{gen}}$4 and $S_{\mathrm{gen}}$5 to the semiclassical region far from the singularity. The paper’s conclusion is that these QES are “always driven to the semiclassical region far from the singularity,” and closed cosmologies of this type do not produce islands (Manu et al., 2020).

“Cosmologies, singularities and quantum extremal surfaces” sharpens this by introducing a spatial regulator $S_{\mathrm{gen}}$6, interpreted as a stretched boundary. In isotropic AdS Kasner the regulated QES time satisfies

$S_{\mathrm{gen}}$7

and for $S_{\mathrm{gen}}$8,

$S_{\mathrm{gen}}$9

The QES therefore lags behind the observer location, again away from the singularity. The same analysis finds that a potential island-like region becomes inconsistent when examined near the putative island boundary. In null Kasner backgrounds, by contrast, the QES can move toward the near-singularity region, although the on-shell generalized entropy is then generically singular (Goswami et al., 2021).

De Sitter and Schwarzschild–de Sitter studies broaden the meaning of stretching further. “Schwarzschild de Sitter and extremal surfaces” finds codimension-2 timelike extremal surfaces stretching between $*$0 and $*$1, approaching the cosmological horizon and becoming nearly null there, as well as spacelike surfaces that stretch indefinitely across the extended Penrose diagram and can pass near both cosmological and Schwarzschild horizons. The paper does not compute $*$2, but explicitly presents these classical surfaces as the geometric backbone on which stretched QES might later be constructed (Fernandes et al., 2019). “de Sitter space, extremal surfaces and ‘time-entanglement’” likewise studies future-past timelike extremal surfaces in Lorentzian de Sitter and no-boundary continuations that join a timelike top half to a Euclidean hemisphere. The areas are purely imaginary in the fully Lorentzian case and complex in the no-boundary case, which implies that any generalized entropy built from them could also be complex. Here too the stretched-QES language is prospective rather than fully realized by an explicit $*$3 calculation (Narayan, 2022).

6. Exactness, phase transitions, and conceptual status

One source of ambiguity is that “stretching” may refer either to a geometric surface or to a transition region in entropic dominance. “The refined quantum extremal surface prescription from the asymptotic equipartition property” replaces the sharp two-saddle von Neumann transition by a one-shot-entropic window bounded by smooth min- and max-entropies. For two competing saddles with areas $*$4 and $*$5,

$*$6

defines an indefinite region in which generalized entropy need not equal the boundary entropy. This is a stretched QES transition rather than a stretched geometric surface. The same paper proves that for fixed-area states with pure bulk marginals, higher Rényi entropies retain sharp transitions, so the broadening is specific to the $*$7 von Neumann limit (Wang, 2021).

At the level of exact wedge boundaries, the literature is restrictive. Engelhardt and Wall proved that QES lie outside the causal wedge and obey barrier theorems: quantum trapped or quantum marginally trapped null surfaces obstruct QES from entering certain regions, with implications for bulk reconstruction (Engelhardt et al., 2014). Gao’s operator-based argument in JT gravity strengthens this: if a candidate boundary is not exactly quantum extremal, then a complementary Connes cocycle flow can violate complementary causal wedge exclusion. The relevant threshold is finite whenever the outward quantum expansion is positive,

$*$8

This means that even an arbitrarily small but nonzero departure from extremality can become inconsistent under sufficiently strong complementary modular flow. A plausible implication is that stretched QES are admissible as regulators, perturbative proxies, operational diagnostics, or algebraic shells, but not as literal replacements for the exact entanglement wedge boundary without additional error-control assumptions (Soni, 2024).

Taken together, the literature supports a layered picture. Exact entanglement wedge boundaries remain exact QES. Around them, however, several technically useful stretched notions appear: perturbatively displaced surfaces solving elliptic deviation equations, horizon-proximate semi-classical saddles in JT gravity, operationally smeared stringy targets, algebraically thick shells generated by split failure, discretized row-localized tensor-network surfaces, and regulator-supported or future-past surfaces in cosmological settings. The unifying theme is not a universal new entropy prescription, but a family of controlled refinements that expose how generalized entropy, modular structure, string nonlocality, and semiclassical backreaction blur the idealized sharpness of extremal surfaces in different regimes (Engelhardt et al., 2019, Pedraza et al., 2021, Herderschee et al., 2 Oct 2025).