Quantum Metric and Entanglement on Spin Networks

TL;DR

This paper explores how geometric quantum mechanics can characterize entanglement in spin network states, linking quantum entanglement measures to geometric properties like surface area in quantum gravity models.

Contribution

It introduces a tensorial framework using GQM to analyze entanglement on spin networks, connecting entanglement measures to geometric features such as surface area.

Findings

Entanglement monotone acts as a distance measure for separability.

Maximally entangled states relate entanglement to surface area.

Tensorial characterization encodes separability and entanglement information.

Abstract

Motivated by the idea that, in the background-independent framework of a Quantum Theory of Gravity, entanglement is expected to play a key role in the reconstruction of spacetime geometry, we investigate the possibility of using the formalism of Geometric Quantum Mechanics (GQM) to give a tensorial characterization of entanglement on spin network states. Our analysis focuses on the simple case of a single link graph (Wilson line state) for which we define a dictionary to construct a Riemannian metric tensor and a symplectic structure on the space of states. The manifold of (pure) quantum states is then stratified in terms of orbits of equally entangled states and the block-coefficient matrices of the corresponding pulled-back tensors fully encode the information about separability and entanglement. In particular, the off-diagonal blocks define an entanglement monotone interpreted as a distance with respect to the separable state. As such, it provides a measure of graph connectivity. Finally, 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. This suggests a connection between the GQM formalism and the (simplicial) geometric properties of spin network states through entanglement.