# Entanglement as geometry and flow

## TL;DR

This paper investigates how entanglement in quantum systems can be represented geometrically through an adjacency matrix, linking entanglement measures to geometry and proposing a new perspective on entanglement flow and metrics.

## Contribution

It introduces a generalized adjacency matrix representing all bipartitions' entanglement, connecting it to mutual information and correlators, and proposes a geometric interpretation of entanglement.

## Key findings

- Adjacency matrix elements match mutual information in exact cases.
- Provides a good approximation of entanglement structure in general.
- Suggests a metric derived from entanglement for one-dimensional conformal systems.

## Abstract

We explore the connection between the area law for entanglement and geometry by representing the entanglement entropies corresponding to all $2^N$ bipartitions of an $N$-party pure quantum system by means of a (generalized) adjacency matrix. In the cases where the representation is exact, the elements of that matrix coincide with the mutual information between pairs of sites. In others, it provides a very good approximation, and in all the cases it yields a natural {\em entanglement contour} which is similar to previous proposals.Moreover, for one-dimensional conformal invariant systems, the generalized adjacency matrix is given by the two-point correlator of an {\em entanglement current} operator. We conjecture how this entanglement current may give rise to a metric entirely built from entanglement.