Gluing polyhedra with entanglement in loop quantum gravity

In a spin-network basis state, nodes of the graph describe unentangled quantum regions of space, quantum polyhedra. In this paper we show how entanglement between intertwiner degrees of freedom enforces gluing conditions for neighboring quantum polyhedra. In particular, we introduce Bell-network states, entangled states defined via squeezed vacuum techniques. We study correlations of quantum polyhedra in a dipole, a pentagram, and a generic graph. We find that vector geometries, structures with neighboring polyhedra having adjacent faces glued back to back, arise from Bell-network states. We also discuss the relation to Regge geometries. The results presented show clearly the role that entanglement plays in the gluing of neighboring quantum regions of space.