# Fisher Metric, Geometric Entanglement and Spin Networks

## TL;DR

This paper introduces an information geometric approach to quantify entanglement in spin network states, linking quantum correlations to geometric properties in quantum gravity.

## Contribution

It develops a Fisher metric-based entanglement measure for spin networks, connecting entanglement with geometric features like surface area in quantum gravity.

## Key findings

- Fisher metric defines an entanglement monotone proportional to surface area.
- Maximally entangled states correspond to geometric surfaces in spin networks.
- Entanglement encodes both local and non-local quantum correlations in spin networks.

## Abstract

Starting from recent results on the geometric formulation of quantum mechanics, we propose a new information geometric characterization of entanglement for spin network states in the context of quantum gravity. For the simple case of a single-link fixed graph (Wilson line), we detail the construction of a Riemannian Fisher metric tensor and a symplectic structure on the graph Hilbert space, showing how these encode the whole information about separability and entanglement. In particular, the Fisher metric defines an entanglement monotone which provides a notion of distance among states in the Hilbert space. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We further extend such analysis to the study of non-local correlations between two non-adjacent regions of a generic spin network graph characterized by the bipartite unfolding of an Intertwiner state. Our analysis confirms the interpretation of spin network bonds as a result of entanglement and to regard the same spin network graph as an information graph, whose connectivity encodes, both at the local and non-local level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.