# Geometry of Entanglement

## TL;DR

This paper introduces a geometric measure based on geodesic curvature to quantify bipartite entanglement in dual CFT states, connecting geometric and tensor network approaches.

## Contribution

It proposes a novel geometric measure for entanglement using geodesic curvature within the surface-state correspondence framework.

## Key findings

- Quantitative agreement with MERA tensor network calculations
- Application of Gauss-Bonnet theorem to entanglement measures
- New geometric interpretation of bipartite entanglement

## Abstract

In the context of the surface-state correspondence we propose the geodesic curvature of a convex curve as a local measure of factorization of the dual CFT state. Its integral will be interpreted as computing the bipartite entanglement among degrees of freedom with support on the chosen domain. We will derive results through application of the Gauss-Bonnet theorem and show quantitative agreement with computations using the MERA tensor network and the formalism of entanglement density.