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

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