Effective geometry of Bell-network states on a dipole graph

TL;DR

This paper analyzes the effective quantum geometry of Bell-network states on a dipole graph in loop quantum gravity, revealing that the average geometry resembles spherical tetrahedra with significant fluctuations and correlations.

Contribution

It provides a detailed characterization of the quantum geometry of Bell-network states on a dipole graph, highlighting their spherical tetrahedral nature and correlation properties.

Findings

Average geometry resembles spherical tetrahedra

Significant fluctuations in dihedral angles

Strong correlations between nodes

Abstract

Bell-network states are a class of entangled states of the geometry that satisfy an area-law for the entanglement entropy in a limit of large spins and are automorphism-invariant, for arbitrary graphs. We present a comprehensive analysis of the effective geometry of Bell-network states on a dipole graph. Our main goal is to provide a detailed characterization of the quantum geometry of a class of diffeomorphism-invariant, area-law states representing homogeneous and isotropic configurations in loop quantum gravity, which may be explored as boundary states for the dynamics of the theory. We found that the average geometry at each node in the dipole graph does not match that of a flat tetrahedron. Instead, the expected values of the geometric observables satisfy relations that are characteristic of spherical tetrahedra. The mean geometry is accompanied by fluctuations with considerable relative dispersion for the dihedral angle, and perfectly correlated for the two nodes.