The entanglement wedge polygon

A bstract In this work we consider a particular codimension-one 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 A i the EWP is defined as the region with maximal volume external to all the individual homology regions $$ {r}_{A_i} $$ r A i which consist of the intersection of the entanglement wedge EW( A i ) with the time slice. In three dimensions, when the EWP has constant curvature as in vacuum AdS 3 and BTZ spacetime, its volume is topological as a direct consequence of the Gauss-Bonnet theorem. The topological nature is lost in the time-dependent covariant case and in the presence of an end-of-the-world brane. In higher dimensions we make progress by considering a number of concrete examples including vacuum, black brane, and soliton solutions of AdS 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.