Holographic geometry/real-space entanglement correspondence and metric reconstruction

In holography, the boundary entanglement structure is believed to be encoded in the bulk geometry. In this work, we investigate the precise correspondence between the boundary real-space entanglement and the bulk geometry. By the boundary real-space entanglement, we refer to the conditional mutual information (CMI) for two infinitesimal subsystems separated by a distance $l$, and the corresponding bulk geometry is at a radial position $z_{\ast }$, namely the turning point of the entanglement wedge for a boundary region with a length scale $l$. In a generic geometry described by a given coordinate system, $z_{\ast }$ can be determined locally by $l$, while the exact expression for $z_{\ast }(l)$ depends on the gauge choice, reflecting the inherent nonlocality of this seemingly local correspondence. We propose to specify the function $z_{\ast }(l)$ as the criterion for a gauge choice, and with the specified gauge function, we verify the exact correspondence between the boundary real-space entanglement and the bulk geometry. Inspired by this correspondence, we propose a new method of bulk metric reconstruction from boundary entanglement data, namely the CMI reconstruction. In this CMI proposal, with the gauge fixed a priori by specifying $z_{\ast }(l)$, the bulk metric can be reconstructed from the relation between the bulk geometry and the boundary CMI. The CMI reconstruction method establishes a connection between the differential entropy prescription and Bilson’s general algorithm for metric reconstruction.