Local geometry from entanglement entropy

TL;DR

This paper demonstrates how to directly derive the AdS$_3$ geometry in Riemann Normal Coordinates from the entanglement entropy of 2D conformal field theories, simplifying the connection between quantum entanglement and spacetime geometry.

Contribution

It provides a method to extract the AdS$_3$ metric in RNC directly from entanglement entropy in CFT$_2$, advancing the 'it from qubit' program.

Findings

AdS$_3$ metric in RNC can be obtained from CFT$_2$ entanglement entropy.

Finite temperature and length CFT$_2$ examples validate the approach.

The method simplifies the reconstruction of bulk geometry from boundary entanglement data.

Abstract

Constructing the corresponding geometries from given entanglement entropies of a boundary QFT is a big challenge and leads to the grand project \emph{ it from Qubit}. Based on the observation that the AdS metric in the Riemann Normal Coordinates (RNC) can be summed into a closed form, we find that the AdS$_3$ metric in RNC can be straightforwardly read off from the entanglement entropy of CFT$_2$. We use the finite length or finite temperature CFT$_2$ as examples to demonstrate the identification.