Hyperinvariant spin network states — an AdS/CFT model from first principles

We study the existence and limitations of hyperinvariant tensor networks incorporating a local SU(2) symmetry. As discrete implementations of the anti de-Sitter/conformal field theory (AdS/CFT) correspondence, such networks have created bridges between the fields of quantum information theory and quantum gravity. Adding SU(2) symmetry to the tensor network allows a direct connection to spin network states, a basis of the kinematic Hilbert space of loop quantum gravity (LQG). We consider a particular situation where the states can be interpreted as kinematic quantum states for three-dimensional quantum gravity. We demonstrate an entanglement-geometry correspondence realized in certain quantum states of the gravitational field in LQG, showing that models similar to the ones introduced by [F. Pastawski et al., JHEP 06, 149 (2015)] can be obtained as quantum gravitational models from first principles. We provide examples of hyperinvariant tensor networks, but also prove constraints on their existence in the form of no-go theorems that exclude absolutely maximally entangled states as well as general holographic codes from local SU(2)-invariance, as long as the logical index also transforms covariantly under the gauge group. We calculate surface areas as expectation values of the LQG area operator and discuss further possible constraints as a consequence of a decay of correlations on the boundary.