Weaving the (AdS) spaces with partial entanglement entropy threads

In the context of the AdS/CFT correspondence, we propose a general scheme for reconstructing bulk geometric quantities in a static pure AdS background using the partial entanglement entropy (PEE), a measure of the entanglement structure on the boundary CFT. The PEE between any two points $\mathcal{I}(\vec{x}, \vec{y})$ serves as the fundamental building block of the PEE structure. Any two-point PEE $\mathcal{I}(\vec{x}, \vec{y})$ can be geometrized by the bulk geodesic connecting two boundary points $\vec{x}$ and $\vec{y}$, which we call the PEE thread, with the density of the threads determined by the boundary PEE structure. In the AdS bulk, the set of all the PEE threads forms a continuous ``network'', which we call the PEE network. In this paper, we show that the density of the PEE threads passing through any bulk point is exactly $1/(4G)$. Based on this observation we give a reformulation of the Ryu-Takayanagi (RT) formula for a generic boundary region in general dimensional Poincar\'e AdS space. More explicitly, for any static boundary region $A$, the homologous surface $\Sigma_{A}$ that has the minimal number of intersections with the bulk PEE network is exactly the RT surface of $A$, and the minimal number of intersections reproduces the holographic entanglement entropy. The reconstruction for the area of bulk geometric quantities by counting the number of intersections with the bulk PEE network applies to generic bulk geometric quantities. Interestingly, this reconstruction indicates a pure geometric statement, which is exactly the so-called \emph{Crofton formula} in Poincar\'e AdS.