Generalized Entanglement Wedges in Holography

Updated 4 July 2026

Generalized entanglement wedges are extended constructs in holography that assign bulk regions to gravitating domains via generalized entropy measures.
They unify static and covariant formulations by distinguishing between max- and min-wedges to overcome limitations of standard entanglement wedge reconstruction.
Applications include enhanced bulk reconstruction, operational access through simple wedges, and the extension of holographic entropy inequalities beyond AdS/CFT.

Generalized entanglement wedges are extensions of the entanglement-wedge concept beyond the standard assignment of a boundary subregion to a bulk reconstructible region in AdS/CFT. In one influential formulation, an arbitrary gravitating region $a$ is assigned a bulk region $E(a)\supset a$ on a time-reflection-symmetric slice, or more generally a pair of covariant regions $e_{\max}(a)$ and $e_{\min}(a)$ , intended to quantify the range of holographic encoding in general spacetimes (Bousso et al., 2022, Bousso et al., 2023). A distinct but related line of work uses the language of generalized entanglement wedges to enlarge the data required for reconstruction inside an ordinary wedge, notably when differential entropy fails because of “entanglement shade” (Espíndola et al., 2018). Across these usages, the common theme is that reconstructibility in gravity is not exhausted by boundary-anchored HRT surfaces and often depends on generalized entropy, auxiliary purifier data, causal structure, or observer-dependent access.

1. Historical formation and conceptual scope

The immediate precursor to later generalized-wedge frameworks was the study of reconstruction inside an ordinary entanglement wedge for extended bulk objects. In AdS $_3$ Rindler space, it was shown that a generic spacelike curve inside a boundary region’s entanglement wedge is not fully reconstructible from interval entanglement entropies, even with the covariant “null alignment” generalization of hole-ography. The obstruction is not merely a forbidden bulk region but a forbidden set of point-and-slope data in the tangent bundle, termed the entanglement shade (Espíndola et al., 2018). In that setting, ordinary differential entropy reconstructs only the unshaded segments of a curve, while the missing segments require a generalized entanglement-of-purification observable and its associated “differential purification.” The result is a complete reconstruction of all spacelike curves within an arbitrary entanglement wedge in any three-dimensional bulk geometry (Espíndola et al., 2018).

A more sweeping generalization emerged from the attempt to assign entanglement wedges directly to gravitating bulk regions, rather than only to asymptotic or nongravitating subsystems. The central motivation was that Hawking radiation in a weakly gravitating bulk region, or a near-boundary AdS bulk region, should already encode regions that standard discussions attach to nongravitating auxiliary systems or boundary subregions. This led to a bulk-to-bulk map

$a \longmapsto E(a)\supset a,$

where $E(a)$ is meant to characterize the spatial range of holographic encoding for the full algebra generated by quasi-local operators in $a$ (Bousso et al., 2022).

The covariant reformulation sharpened this picture by separating two logically distinct notions: a largest region whose information can flow inward to $a$ , and a smallest region outside of which information can flow outward away from $a$ . These become $E(a)\supset a$ 0 and $E(a)\supset a$ 1, and unlike their AdS/CFT analogues they can differ already at the classical level, even when generalized entropy is approximated by area (Bousso et al., 2023). This distinction is one of the characteristic features of generalized entanglement wedges in arbitrary spacetimes.

2. Static generalized entanglement wedges for gravitating regions

The cleanest formulation is static. On a time-reflection-symmetric Cauchy slice $E(a)\supset a$ 2, an open subset $E(a)\supset a$ 3 is treated as a wedge. Its generalized entropy is

$E(a)\supset a$ 4

with $E(a)\supset a$ 5 and the ellipsis denoting local counterterms (Bousso et al., 2022). The generalized entanglement wedge $E(a)\supset a$ 6 is then defined as the wedge satisfying

$E(a)\supset a$ 7

and having the smallest generalized entropy among all such wedges (Bousso et al., 2022).

This definition is bulk-intrinsic. If $E(a)\supset a$ 8 is compact, the minimization is over all supersets of $E(a)\supset a$ 9; if $e_{\max}(a)$ 0 reaches the conformal boundary, the asymptotic footprint is fixed. In the classical limit, when matter entropy is negligible, $e_{\max}(a)$ 1 reduces to an area-minimizing superset; in flat-space examples discussed in the proposal, this becomes the convex hull of a nonconvex region (Bousso et al., 2022).

A key consistency check is reduction to the ordinary entanglement wedge. If $e_{\max}(a)$ 2 lies inside the standard wedge of its conformal boundary, then

$e_{\max}(a)$ 3

so the construction reproduces the usual static QES prescription near the AdS boundary (Bousso et al., 2022).

The static theory also obeys several structural properties expected of an encoding map. If $e_{\max}(a)$ 4, then the minimized generalized entropies satisfy monotonicity, and the regions themselves satisfy nesting,

$e_{\max}(a)$ 5

There is a no-cloning statement: if $e_{\max}(a)$ 6 and $e_{\max}(a)$ 7, then $e_{\max}(a)$ 8. There is also a strong-subadditivity analogue,

$e_{\max}(a)$ 9

for mutually disjoint open subsets $e_{\min}(a)$ 0 (Bousso et al., 2022). These results are among the main reasons generalized entanglement wedges were proposed as a candidate universal feature of quantum gravity rather than an AdS-specific artifact.

3. Covariant max- and min-entanglement wedges

The covariant framework replaces a single wedge by two. In a globally hyperbolic spacetime $e_{\min}(a)$ 1, the spacelike complement of a set $e_{\min}(a)$ 2 is

$e_{\min}(a)$ 3

and a wedge is a set satisfying

$e_{\min}(a)$ 4

The edge of a wedge is

$e_{\min}(a)$ 5

and generalized entropy takes the form

$e_{\min}(a)$ 6

(Bousso et al., 2023).

The construction then distinguishes $e_{\min}(a)$ 7 and $e_{\min}(a)$ 8. The max-entanglement wedge is the wedge union of all admissible supersets of $e_{\min}(a)$ 9 that are antinormal away from the original edge and minimize generalized entropy on a suitable Cauchy slice. The min-entanglement wedge is the intersection of all admissible normal supersets satisfying the complementary minimizing condition. The basic inclusion relation is

$_3$ 0

(Bousso et al., 2023).

Operationally, $_3$ 1 is interpreted as the largest region whose information can be moved inward to $_3$ 2, through quantum channels whose capacity is controlled by generalized entropies of intermediate homology surfaces. By contrast, all information outside $_3$ 3 can flow outward away from $_3$ 4 (Bousso et al., 2023). In time-reflection-symmetric settings these notions collapse: $_3$ 5 recovering the static picture (Bousso et al., 2023).

The covariant formulation is conceptually stronger and technically harder. It is explicitly possible for $_3$ 6 even in the classical area approximation (Bousso et al., 2023). Later work in arbitrary spacetimes therefore often defines a single generalized entanglement wedge $_3$ 7 only when the max and min constructions coincide (Bousso et al., 2024).

A path-integral derivation of the static BP prescription was given using random tensor networks, fixed-geometry states, and a “hollowing” operation that removes the bulk region $_3$ 8 and computes the entropy of the newly opened legs. In time-reflection-symmetric settings this yields

$_3$ 9

with $a \longmapsto E(a)\supset a,$ 0 the smallest generalized-entropy region containing $a \longmapsto E(a)\supset a,$ 1 and sharing its conformal boundary (Kaya et al., 11 Jun 2025). The same paper shows that Rényi saddles can depend on how the bulk region is gauge-invariantly specified, but the $a \longmapsto E(a)\supset a,$ 2 limit universally reproduces the BP prescription (Kaya et al., 11 Jun 2025). A later comparison found that the tensor-network observer rules proposed in a separate context are exactly equivalent to the KRR hollowing rules, suggesting a direct link between observer promotion and generalized entanglement wedges (Bozanic et al., 26 Jun 2026).

4. Entropy inequalities and the generalized holographic entropy cone

One of the strongest developments is the extension of holographic entropy inequalities beyond AdS/CFT. In a framework based on causally defined wedges in arbitrary globally hyperbolic spacetimes, the generalized entanglement wedge $a \longmapsto E(a)\supset a,$ 3 is defined only when a max-entanglement wedge and a min-entanglement wedge coincide. In the regime where matter entropy is negligible compared to geometric entropy, the area of the wedge edge is treated as an entropy-like quantity (Bousso et al., 2024).

Under appropriate independence assumptions, these generalized wedges satisfy monogamy of mutual information: $a \longmapsto E(a)\supset a,$ 4 This extends a genuinely holographic entropy-cone inequality beyond AdS/CFT to arbitrary spacetimes (Bousso et al., 2024).

In the static setting, all known area inequalities of the holographic entropy cone were then extended to generalized entanglement wedges of bulk regions in arbitrary spacetimes. The essential new hypothesis is a mutual independence condition: $a \longmapsto E(a)\supset a,$ 5 which expresses that each input bulk region lies outside the generalized entanglement wedge of the union of all the others (Bousso et al., 5 Feb 2025). Under this condition, every inequality generated by a contraction map in the ordinary static holographic entropy cone carries over to generalized wedges (Bousso et al., 5 Feb 2025).

An algebraic interpretation has been proposed in which each generalized entanglement wedge $a \longmapsto E(a)\supset a,$ 6 is assigned a von Neumann algebra $a \longmapsto E(a)\supset a,$ 7. In that framework, the wedge entropy is conjecturally related to algebraic entropy and index data by

$a \longmapsto E(a)\supset a,$ 8

With suitable assumptions on inclusion, meets and joins, factor properties, and commuting conditional expectations, BP monotonicity and strong subadditivity follow from standard operator-algebraic entropy inequalities (Sahu et al., 26 Nov 2025). This proposal does not construct the microscopic algebras, but it gives a natural algebraic reading of why generalized wedge entropies behave like ordinary quantum entropies.

5. Reconstruction, simple wedges, and operational access

Generalized entanglement wedges also appear in reconstruction problems that refine the ordinary entanglement-wedge paradigm. In AdS $a \longmapsto E(a)\supset a,$ 9 Rindler space, the obstruction called entanglement shade shows that not every tangent direction inside an entanglement wedge corresponds to a boundary-anchored geodesic that remains in the wedge. For the unshaded segments of a spacelike curve, differential entropy reconstructs the length; for shaded segments, the relevant data are nonminimal entanglement wedge cross sections ending on the RT surface, encoded by a generalized entanglement-of-purification observable $E(a)$ 0 and its “differential purification” (Espíndola et al., 2018). In that three-dimensional setting, the full wedge becomes reconstructible only after supplementing $E(a)$ 1-data with auxiliary purifier data naturally associated with the RT surface (Espíndola et al., 2018).

A related operational problem asks how much of an entanglement wedge can be brought into causal contact with the boundary by simple semiclassical operations. For a perturbative class of states, a cocycle-like unitary supported in the causal wedge produces negative null-energy shocks on the causal horizon and moves the causal surface toward the outermost extremal surface. The construction reveals the previously causally inaccessible “peninsula” to first order in its size, up to $E(a)$ 2 accuracy, and is closely related to Connes cocycle flow (Levine et al., 2020).

The general-spacetime analogue of the AdS simple wedge is the simple wedge $E(a)$ 3, defined as the largest wedge accessible from an input wedge $E(a)$ 4 by an alternating zigzag of antinormal lightsheets. The finite zigzag wedges $E(a)$ 5 are accessible from $E(a)$ 6 on explicit preferred Cauchy slices, and the limit $E(a)$ 7 is a throat accessible from $E(a)$ 8, unique, contained in every other accessible throat, and therefore contained in the generalized max-entanglement wedge $E(a)$ 9 (Bousso et al., 1 May 2025). In AdS, with $a$ 0 taken to be the bulk causal wedge of a boundary region, this reproduces the ordinary outermost or simple wedge (Bousso et al., 1 May 2025). This construction isolates the sub-exponentially reconstructible part of the generalized wedge and shows that generalized wedges can contain an internal hierarchy, analogous to the distinction between the simple wedge and the full entanglement wedge in the presence of a Python’s lunch.

Several adjacent constructions broaden the generalized-wedge landscape without being identical to the BP formalism. One is the generalized Rindler wedge (GRW), defined by the region accessible to a family of globally well-defined accelerating observers. Its defining geometric criterion is Rindler-convexity, equivalently that outward normal null geodesics from the boundary of a spatial region never intersect to form caustics, or that any externally tangential lightsphere never reaches the inside of the region (Ju et al., 2023). The associated entropy obeys the same area law,

$a$ 1

and the construction is proposed as an observer-based generalized wedge framework distinct from standard HRT wedges (Ju et al., 2023). A later extension derived separation theorems for squashed entanglement and conditional entanglement of multipartite information in this generalized Rindler-wedge setting, with a time cutoff acting as an operational probe of how quantum correlations disappear under restricted accessibility (Ju et al., 2023). These constructions are best regarded as observer-based analogues and probes of generalized entanglement-wedge ideas rather than replacements for the BP framework.

The generalized-entanglement-wedge formalism has also been used to recast the connected wedge theorem in terms of bulk regions and bulk entropies. For $a$ 2 scattering, new upper and lower bounds on boundary mutual information were obtained in terms of generalized entropies of bulk scattering regions, and new bulk decision regions were defined for which nonempty scattering implies a connected entanglement wedge. The construction extends to asymptotically flat spacetimes, though for the new bulk decision regions the proven connectedness result uses the older $a$ 3 definition rather than the refined $a$ 4 (Arayath et al., 24 Apr 2026).

A different multipartite generalization is the entanglement wedge polygon (EWP), a codimension-one region on a maximal-volume slice obtained by removing the individual homology regions $a$ 5 from the homology region of the union. For pure states,

$a$ 6

and for mixed states the construction is generalized using entanglement wedge cross sections (Fujiki et al., 19 Jun 2026). In vacuum AdS $a$ 7, its volume is topological by Gauss–Bonnet: $a$ 8 The EWP is not a generalized entanglement wedge in the BP sense, but it is a codimension-one residual region built from ordinary wedges and wedge cross sections, aimed at isolating multipartite bulk structure (Fujiki et al., 19 Jun 2026).

The subject remains open in several directions. Higher-dimensional analogues of differential purification were left as future work already in the AdS $a$ 9 reconstruction program (Espíndola et al., 2018). The static BP proposal is much more mature than the fully covariant story; no covariant definition is known that simultaneously reproduces the static wedge, reduces to the standard boundary entanglement wedge, and preserves all desired properties such as nesting and strong subadditivity (Bousso et al., 2022). In arbitrary spacetimes, a single generalized wedge $a$ 0 may fail to exist because the max and min constructions need not coincide (Bousso et al., 2024). At the microscopic level, the hypothesized algebras associated with generalized wedges remain largely unknown (Sahu et al., 26 Nov 2025). These limitations are structural rather than incidental: they indicate that generalized entanglement wedges are not merely an extrapolation of HRT surfaces, but a broader program for formulating holographic encoding, entropy, and observer access in gravitating systems beyond the standard boundary-subregion setting.