Holographic geometry/real-space entanglement correspondence and metric reconstruction

Abstract: 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 $z_$, 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, $z_$ can be determined locally by $l$, while the exact expression for $z_(l)$ depends on the gauge choice, reflecting the inherent nonlocality of this seemingly local correspondence. We propose to specify the function $z_(l)$ 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 $z_*(l)$, 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.