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
n-to-
n connected wedge theorem, which considers
n input and
n output locations at the boundary of an asymptotically AdS
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
n 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
n-to-
n 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
2-to-
2 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.
We investigate the geometry of entanglement wedges for asymptotic
n-to-
n scattering configurations in asymptotically AdS
3 spacetimes. Extending the
2-to-
2 Connected Wedge Theorem, we establish a strictly weaker sufficient condition for the input entanglement wedge
E(V1∪⋯∪Vn) to be connected: the existence of a single pair of input regions satisfying a
2-to-all causal intersection condition already forces full multipartite wedge connectivity. We also derive novel necessary conditions, showing that when the input wedge is connected, certain output ridges must enter the input wedge, and we organize these consequences into a layered reduction on the boundary lattice. Furthermore, we analyze the generalized bulk scattering region
SE=E(V1∪⋯∪Vn)∩E(W1∪⋯∪Wn) and obtain necessary conditions for it to be nonempty; for
n>2 these conditions are stronger than mere wedge connectedness. Our results provide new geometric restrictions on multipartite entanglement in holography and clarify the holographic dictionary for multi-partite scattering processes, while also highlighting intrinsic limitations for
n>2.
A bstract We investigate the geometry of entanglement wedges for asymptotic n -to- n scattering configurations in asymptotically AdS 3 spacetimes. Extending the 2-to-2 Connected Wedge Theorem, we establish a strictly weaker sufficient condition for the input entanglement wedge
E(V1∪⋯∪Vn) E V 1 ∪ ⋯ ∪ V n to be connected: the existence of a single pair of input regions satisfying a 2-to-all causal intersection condition already forces full multipartite wedge connectivity. We also derive novel necessary conditions, showing that when the input wedge is connected, certain output ridges must enter the input wedge, and we organize these consequences into a layered reduction on the boundary lattice. Furthermore, we analyze the generalized bulk scattering region
SE=E(V1∪⋯∪Vn)∩E(W1∪⋯∪Wn) S E = E V 1 ∪ ⋯ ∪ V n ∩ E W 1 ∪ ⋯ ∪ W n and obtain necessary conditions for it to be nonempty; for n > 2 these conditions are stronger than mere wedge connectedness. Our results provide new geometric restrictions on multipartite entanglement in holography and clarify the holographic dictionary for multi-partite scattering processes, while also highlighting intrinsic limitations for n > 2.
A bstract 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 n -to- n connected wedge theorem, which considers n input and n output locations at the boundary of an asymptotically AdS 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 n 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 n -to- n 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 2-to-2 connected wedge theorem proven in [3]. We also correct some errors in the proof presented there, in particular a false claim that existing proof techniques work above three bulk dimensions.
A bstract In AdS/CFT, there can exist local 2-to-2 bulk scattering processes even when local scattering is not possible on the boundary; these have previously been studied in con- nection with boundary correlation functions. We show that boundary regions associated with these scattering configurations must have O (1 /G N ) mutual information, and hence a connected entanglement wedge. One of us previously argued for this statement from the boundary theory using operational tools in quantum information theory. We improve that argument to make it robust to small errors and provide a proof in the bulk using focusing arguments in general relativity. We also provide a direct link to entanglement wedge reconstruction by showing that the bulk scattering region must lie inside the con- nected entanglement wedge. Our construction implies the existence of nonlocal quantum computation protocols that are exponentially more efficient than the optimal protocols currently known.
In AdS/CFT, there can exist local 2-to-2 bulk scattering processes even when local scattering is not possible on the boundary; these have previously been studied in con- nection with boundary correlation functions. We show that boundary regions associated with these scattering configurations must have O(1/GN) mutual information, and hence a connected entanglement wedge. One of us previously argued for this statement from the boundary theory using operational tools in quantum information theory. We improve that argument to make it robust to small errors and provide a proof in the bulk using focusing arguments in general relativity. We also provide a direct link to entanglement wedge reconstruction by showing that the bulk scattering region must lie inside the con- nected entanglement wedge. Our construction implies the existence of nonlocal quantum computation protocols that are exponentially more efficient than the optimal protocols currently known.