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arXiv:2610.07176

Generalised entanglement wedges in three dimensions

Abhisek Sahu

hep-thquant-phSubmitted Oct 5, 2026• Announced Oct 7, 2026• 0 citations
v1 · Oct 5, 2026v2 · Oct 14, 2026
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Abstract

Generalized entanglement wedges (GWs), proposed by Bousso and Penington for gravitating regions in general spacetimes, share many properties of ordinary entanglement wedges, including nesting, monotonicity and strong subadditivity of their generalized entropies. In AdS/CFT, the corresponding properties of entanglement wedges are consequences of the structure of the dual CFT. In this paper we ask whether the properties of GWs have a similar origin. We explore this question in vacuum AdS$_3$ at leading order in $G$. We first show that every GW on a static slice is a union of geodesically convex regions. Each such region is an intersection of ordinary entanglement wedges, so every GW is built from unions and intersections of entanglement wedges. We then use the Crofton formula of integral geometry to write the perimeter of a convex region as the measure of the set of geodesics that intersect it. In AdS$_3$ every geodesic is the Ryu--Takayanagi surface of a boundary interval, so this measure can be written in terms of the von Neumann entropies of CFT intervals. Strong subadditivity of the CFT entropy guarantees that the measure is positive. Monotonicity and strong subadditivity of GWs then follow from simple inequalities between sets of geodesics. The strong subadditivity of GWs is therefore a direct consequence of strong subadditivity on the boundary. Our results further suggest that GWs correspond to subsystems of the CFT that are more general than boundary subregions.

Comments
11 pages, 3 figures

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Generalized entanglement wedges (GWs), proposed by Bousso and Penington for gravitating regions in general spacetimes, share many properties of ordinary entanglement wedges, including nesting, monotonicity and strong subadditivity of their generalized entropies. In AdS/CFT, the corresponding properties of entanglement wedges are consequences of the structure of the dual CFT. In this paper we ask whether the properties of GWs have a similar origin. We explore this question in vacuum AdS3_33​ at leading order in GGG. We first show that every GW on a static slice is a union of geodesically convex regions. Each such region is an intersection of ordinary entanglement wedges, so every GW is built from unions and intersections of entanglement wedges. We then use the Crofton formula of integral geometry to write the perimeter of a convex region as the measure of the set of geodesics that intersect it. In AdS3_33​ every geodesic is the Ryu--Takayanagi surface of a boundary interval, so this measure can be written in terms of the von Neumann entropies of CFT intervals. Strong subadditivity of the CFT entropy guarantees that the measure is positive. Monotonicity and strong subadditivity of GWs then follow from simple inequalities between sets of geodesics. The strong subadditivity of GWs is therefore a direct consequence of strong subadditivity on the boundary. Our results further suggest that GWs correspond to subsystems of the CFT that are more general than boundary subregions.
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A bstract The concept of the generalized entanglement wedge was recently proposed by Bousso and Penington, which states that any bulk gravitational region a possesses an associated generalized entanglement wedge E ( a ) ⊃ a on a static Cauchy surface M in general gravitational spacetimes, where E ( a ) may contain an entanglement island I ( a ). It suggests that the fine-grained entropy for bulk region a is given by the generalized entropy S gen ( E ( a )). Motivated by this proposal, we extend the quantum bit thread description to general gravitational spacetimes, no longer limited to the AdS spacetime. By utilizing the convex optimization techniques, a dual flow description for the generalized entropy S gen ( E ( a )) of a bulk gravitational region a is established on the static Cauchy surface M , such that S gen ( E ( a )) is equal to the maximum flux of any flow that starts from the boundary ∂M and ends at bulk region a , or equivalently, the maximum number of bit threads that connect the boundary ∂M to the bulk region a . In addition, the nesting property of flows is also proved. Thus the basic properties of the entropy for bulk regions, i.e. the monotonicity, subadditivity, Araki-Lieb inequality and strong subadditivity, can be verified from flow perspectives by using properties of flows, such as the nesting property. Moreover, in max thread configurations, we find that there exists some lower bounds on the bulk entanglement entropy of matter fields in the region E ( a ) \ a , particularly on an entanglement island region I ( a ) ⊂ ( E ( a ) \ a ), as required by the existence of a nontrivial generalized entanglement wedge. Our quantum bit thread formulation may provide a way to investigate more fine-grained entanglement structures in general spacetimes.
In this work we consider a particular codimension-1 region of a holographic spacetime which we call the entanglement wedge polygon (EWP). For a pure state and a partition of the boundary into a number of regions AiA_iAi​ the EWP is defined as the region external to all the individual homology regions rAir_{A_i}rAi​​ which consists of the intersection of the entanglement wedge EW(AiA_iAi​) with the time slice. In vacuum AdS3_33​ and BTZ spacetime, the quantity is topological as a direct consequence of the Gauss-Bonnet theorem. In higher dimensions we make progress by considering a number of concrete examples including vacuum, black brane, and soliton solutions of AdSd+1_{d+1}d+1​ as well as spacetime geometries with end of the world branes dual to boundary conformal field theories. We provide a suitable generalization to mixed states and comment on possible connections between the EWP and measures of multi-partite entanglement.
We define a generalized entanglement measure in the context of AdS3_33​/CFT2_22​. Compared to the ordinary entanglement entropy between spatial degrees of freedom dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary CFT. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including regions known as entanglement shadows which are not reached by any Ryu-Takayanagi surface. This allows us to describe regions close to the black hole horizon in the one-sided black hole and the wormhole growth in the case of a two-sided black hole from entanglement data in the boundary field theory, leading us to propose that entanglement is in fact enough to reconstruct the full BTZ geometry from boundary data.
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In the AdS/CFT correspondence, bulk causal structure has consequences for boundary entanglement. In quantum information science, causal structures can be replaced by distributed entanglement for the purposes of information processing. In this work, we deepen the understanding of both of these statements, and their relationship, with a number of new results. Centrally, we present and prove a new theorem, the nnn-to-nnn connected wedge theorem, which considers nnn input and nnn output locations at the boundary of an asymptotically AdS2+1_{2+1}2+1​ spacetime described by AdS/CFT. When a sufficiently strong set of causal connections exists among these points in the bulk, a set of nnn associated regions in the boundary will have extensive-in-N mutual information across any bipartition of the regions. The proof holds in three bulk dimensions for classical spacetimes satisfying the null curvature condition and for semiclassical spacetimes satisfying standard conjectures. The nnn-to-nnn connected wedge theorem gives a precise example of how causal connections in a bulk state can emerge from large-N entanglement features of its boundary dual. It also has consequences for quantum information theory: it reveals one pattern of entanglement which is sufficient for information processing in a particular class of causal networks. We argue this pattern is also necessary, and give an AdS/CFT inspired protocol for information processing in this setting. Our theorem generalizes the 222-to-222 connected wedge theorem proven in arXiv:1912.05649. We also correct some errors in the proof presented there, in particular a false claim that existing proof techniques work above three bulk dimensions.