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.

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.