Information Wedge in Holography

• Information Wedge is a bulk region defined by quantum extremal surfaces that encode entanglement entropy, reconstructability, and boundary correlations in holographic theories.
• It underpins models in AdS/CFT by linking entanglement wedges, causal wedges, and scattering tasks to diagnose mutual information and bulk connectivity.
• Refined measures such as the entanglement wedge cross section and EWCS triangle information provide deeper insights into multipartite quantum correlations and tensor-network reconstructions.

In contemporary high-energy and quantum-information literature, an information wedge most commonly denotes the bulk region encoded in a specified subsystem, usually identified with the entanglement wedge and, in quantum settings, with the region bounded by a quantum extremal surface. In AdS/CFT this notion links reconstructability, entropy, and geometry: the wedge associated with a boundary region can be characterized by Ryu–Takayanagi or Hubeny–Rangamani–Takayanagi surfaces, by generalized entropy, and by operational statements about which bulk processes are visible from boundary correlations (Penington, 2019, Zhao, 27 Sep 2025). Subsequent work has extended the concept from boundary subregions to arbitrary bulk regions, related wedge connectivity to scattering tasks and mutual information, and introduced refined correlation measures that probe internal structure of the wedge rather than only its boundary area (Arayath et al., 24 Apr 2026, Umemoto, 2019).

1. Core holographic meaning

Given a boundary region $B$, the entanglement wedge is the bulk domain of dependence bounded by $B$ and its RT surface $\chi_B$. In the quantum-corrected setting, $\chi_B$ is the quantum extremal surface that extremizes and minimizes

$$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$

and the wedge is the bulk region between $B$ and $\chi_B$ (Penington, 2019). In the formulation used for asymptotic scattering in AdS/CFT, the same reconstructable region is denoted $E(A)$ for a boundary region $A$, with standard relations such as $E(A)\cap \partial M=\hat D(A)$ and $B$0 (Zhao, 27 Sep 2025).

A useful distinction is between boundary-defined and bulk-defined wedge constructions.

Notion	Definition	Role
Entanglement wedge $B$1	Bulk region associated with boundary region $B$2	Boundary reconstruction
Causal wedge $B$3	Causally accessible bulk region from $B$4	Geometric lower bound
Generalized entanglement wedge $B$5	Wedges assigned to bulk region $B$6	Bulk-region reconstruction

The bulk generalization is formulated using wedges $B$7 satisfying $B$8, together with spacelike complement, wedge union, and generalized entropy. For a bulk region $B$9, the max- and min-entanglement wedges are defined as

$\chi_B$0

with inclusion, nesting, and complementarity properties such as $\chi_B$1 and $\chi_B$2 (Arayath et al., 24 Apr 2026). In the classical limit used there, one effectively has $\chi_B$3.

2. Connectedness, scattering, and boundary mutual information

A central development is the connected wedge theorem. For a $\chi_B$4 asymptotic scattering setup with input points $\chi_B$5 and output points $\chi_B$6, the boundary decision regions are

$\chi_B$7

$\chi_B$8

while the original bulk scattering region is

$\chi_B$9

The original theorem states that if there is a bulk-only scattering configuration, then the relevant boundary regions must have mutual information of order

$\chi_B$0

and therefore a connected entanglement wedge (May et al., 2019).

This was generalized to an $\chi_B$1-to-$\chi_B$2 theorem in AdS$\chi_B$3. There one defines input regions $\chi_B$4, output regions $\chi_B$5, and a $\chi_B$6-to-all causal graph $\chi_B$7. If $\chi_B$8 is connected, then the entanglement wedge of $\chi_B$9 is connected. In the classical holographic regime this implies extensive mutual information across any bipartition of the input regions, and the paper explicitly corrects earlier claims that the same proof strategy worked above three bulk dimensions (May et al., 2022).

A further refinement introduces the enlarged bulk scattering region

$$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$0

with $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$1. Under the null energy condition, AdS-hyperbolicity, a pure global boundary state, HRRT surfaces found by maximin, and a suitable nonsingular bulk region, $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$2 has positive measure if and only if both $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$3 and $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$4 are connected; if $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$5 is a single point, both wedges are marginal (Zhao, 27 Sep 2025). A related bulk-based reformulation proves upper and lower bounds

$$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$6

and defines bulk decision regions for which nonempty scattering implies a connected generalized entanglement wedge, including in asymptotically flat spacetimes (Arayath et al., 24 Apr 2026).

3. Correlation measures inside the wedge

The internal structure of an information wedge is probed by quantities more refined than entropy or mutual information. The entanglement wedge cross section $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$7 is the minimal RT area of a cut that partitions the entanglement wedge $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$8 into two parts attached to $$S_{\rm gen}(\chi)=\frac{A(\chi)}{4G_N}+S_{\rm bulk}(\chi),$$9 and $B$0, and it obeys

$B$1

A central result is that $B$2 can exceed entanglement measures such as entanglement of formation in holographic regimes, implying that it captures more than purely bipartite quantum entanglement (Umemoto, 2019).

A direct CFT quantity proposed to compute this cross section is the odd entanglement entropy $B$3, defined from the partially transposed density matrix by an odd-replica continuation. The conjectured holographic relation is

$B$4

and explicit AdS$B$5 and planar BTZ calculations reproduce the expected connected and disconnected phases of the entanglement wedge cross section (Tamaoka, 2018).

The same wedge geometry supports additional optimized measures. The entanglement wedge mutual information $B$6, proposed as the holographic dual of the $B$7-correlation, satisfies

$B$8

as well as symmetry, additivity, extensivity, pure-state normalization, and strong superadditivity (Umemoto, 2019). For tripartite mixed states $B$9, the EWCS triangle information is defined by

$\chi_B$0

In the canonical purification it becomes

$\chi_B$1

and it obeys the upper bound

$\chi_B$2

In AdS$\chi_B$3/CFT$\chi_B$4, the maximized quantity vanishes for $\chi_B$5, undergoes a second phase transition at $\chi_B$6, and saturates the assistance bound beyond that point (Ju et al., 25 Dec 2025).

4. Evaporation, decoding, and operational caveats

In evaporating black holes, the information wedge is the entanglement wedge viewed through the inclusion of the Hawking radiation reservoir. Before the Page time, the relevant RT surface for the whole CFT is empty. After the Page time, a nonempty quantum extremal surface appears slightly inside the event horizon, at an infalling time of order the scrambling time into the past, so that part of the interior moves from the CFT wedge to the radiation wedge. The resulting entropy is

$\chi_B$7

which yields the Page curve, and the radiation wedge then supports Hayden–Preskill-type decoding (Penington, 2019).

This reconstruction is not generically state-independent. Immediately after the Page time, interior operators can be reconstructed from the Hawking radiation only if the initial black hole state is known. For a code space of unknown initial states, a single radiation reconstruction requires

$\chi_B$8

and the consistency of this picture depends on nonperturbative reconstruction errors of order $\chi_B$9 (Penington, 2019).

A common misconception is that connected wedge geometry automatically implies operationally distillable EPR entanglement. A contrary proposal argues that, at leading order in holography, one can have

$E(A)$0

with a connected entanglement wedge but vanishing one-shot LO-distillable entanglement,

$E(A)$1

when the minimal surfaces $E(A)$2 are separated in the bulk. In the same framework, one-shot distillation with restricted holographic measurements is instead governed by locally accessible information,

$E(A)$3

with $E(A)$4 defined from the entanglement wedge cross section and the purifier. The connected wedge is therefore a geometric diagnostic of strong correlation, but not by itself a guarantee of nonzero distillable bipartite entanglement at leading order (Mori et al., 2024).

5. Alternative constructions and model studies

The wedge can also be reconstructed from boundary distinguishability data. For locally excited states in two-dimensional holographic CFTs, the Bures metric of reduced density matrices reproduces the AdS$E(A)$5 time-slice metric for a single interval. Outside the entanglement wedge, the metric vanishes; inside, it equals

$E(A)$6

For disconnected intervals, an alternative information metric reproduces the expected connected and disconnected entanglement wedge phases with the correct phase transition up to a very small error. In a free scalar CFT, by contrast, the same construction yields no sharp wedge structure (Suzuki et al., 2019).

Toy tensor-network models make the reconstructability question more combinatorial. Monte Carlo studies comparing the causal wedge, greedy entanglement wedge, minimum entanglement wedge, and a proposed mutual-information wedge conclude that the minimum entanglement wedge,

$E(A)$7

is the best approximation to the true geometric wedge. In the classical RT-size version, the central-tensor inclusion curve becomes a near step function with threshold $E(A)$8 in the holographic pentagon code, the pentagon/hexagon code, and the single-qubit code. The same study rejects the mutual-information wedge as a general reconstruction principle: rises in mutual information do not reliably track inclusion in the geometric or entanglement wedge, and an operator beyond the greedy entanglement wedge can still be reconstructed when the true geometric wedge is larger (Linden, 2024).

6. Other technical meanings of “wedge”

Outside holography, wedge terminology names several distinct structures rather than a single information-theoretic object. In compactly causal symmetric spaces $E(A)$9, wedge domains generalize Rindler wedges. Their defining positivity domain of the modular vector field, the KMS-like analytic domain, and the polar-cone domain coincide, and connected components of these wedges carry standard subspaces satisfying a Bisognano–Wichmann-type property (Neeb et al., 2021). In charged sectors of QED, however, infrared photon dressing makes Lorentz boost generators diverge, so the usual modular localization of photon observables in the Rindler wedge fails (Asorey et al., 2017).

Other literatures use wedge language in still different senses. In celestial soft-theorem constructions, the wedge condition

$A$0

selects the maximal natural domain on which the operator map $A$1 becomes a Lie algebra representation (Charbonnier et al., 8 May 2026). In 21-cm cosmology, astrophysical foregrounds contaminate a wedge-shaped region of $A$2 space with boundary $A$3; wedge filtering removes cosmological modes, but non-Gaussian mode coupling allows recovery of large ionized regions from wedge-filtered images, while dense logarithmically regular baseline layouts can reduce foreground leakage by $A$4–$A$5 orders of magnitude (Gagnon-Hartman et al., 2021, Murray et al., 2018). In accelerator physics, wedge absorbers support reverse emittance exchange, and a QQB dispersion suppressor reduces post-wedge $A$6 from about $A$7 to about $A$8 for a $A$9 muon-collider cooling concept (Karaaslan et al., 24 Jun 2026). In HCI, “Wedge” and “OptWedge” denote off-screen POI guidance figures exploiting amodal completion rather than holographic reconstruction (Miyagawa, 2022). These uses are terminological neighbors, but they are not the holographic information wedge.

Taken together, the modern literature treats the information wedge primarily as a reconstructable region whose geometry encodes entropy, correlation structure, and the feasibility of bulk or boundary information-processing tasks. Its connectedness can diagnose bulk scattering and Page-time interior transfer, its internal cuts support refined correlation measures, and its generalizations move beyond boundary-anchored RT surfaces to wedges assigned directly to bulk regions (Zhao, 27 Sep 2025, Arayath et al., 24 Apr 2026).