Cosmological states in loop quantum gravity on homogeneous graphs

We introduce a class of states characterized by proposed conditions of homogeneity and isotropy in loop quantum gravity and construct concrete examples given by Bell-network states on a special class of homogeneous graphs. Such states provide new representations of cosmological spaces that can be explored for the formulation of cosmological models in the context of loop quantum gravity. We show that their local geometry is described in an automorphism-invariant manner by one-node observables analogous to the one-body observables used in many-body quantum mechanics, and compute the density matrix representing the restriction of global states to the algebra of one-node observables. The von Neumann entropy of this density matrix provides a notion of entanglement entropy of a local region that is invariant under automorphisms and can be applied to states involving superpositions of distinct graphs.

Automated summary

We introduce a class of states characterized by homogeneity and isotropy conditions in loop quantum gravity and provide concrete examples on a special class of homogeneous graphs. These states offer new representations of cosmological spaces for the formulation of cosmological models in loop quantum gravity. We demonstrate that their local geometry is described in an automorphism-invariant manner by one-node observables, and compute the density matrix representing the restriction of global states to one-node observables. The von Neumann entropy of this density matrix provides an invariant measure of entanglement entropy that can be applied to superpositions of distinct graphs.