Abstract
For quantum gravity states associated to open spin network graphs, we study how the entanglement entropy of the boundary degrees of freedom (spins on open edges) is affected by the bulk data, specifically by its combinatorial structure and by the quantum correlations among intertwiner degrees of freedom. For a specific assignment of bulk edge spins and slightly entangled intertwiners, we recover the Ryu-Takayanagi formula (with a properly (discrete) geometric notion of area, thanks to the underlying quantum gravity formalism) and its corrections due to the entanglement entropy of the bulk state. We also show that the presence of a region with highly entangled intertwiners deforms the minimal-area surface, which is then prevented from entering that region when the entanglement entropy of the latter exceeds a certain bound. This entanglement-based mechanism leads thus to the formation of a black hole-like region in the bulk.
Highlights
Entanglement is expected to play a crucial role in the emergence of spacetime and geometry from fundamental quantum entities, and to account for such emergence is a key goal of all quantum gravity approaches
Our work is a generalization of what has been done in the context of random tensor networks by Hayden et al in [31], with a crucial change in perspective: the tensor networks we work with inherently possess a quantum-geometry characterization, being dual by construction to a triangulation of quantum space
For spin network states corresponding to random tensor networks, how the Renyi-2 entropy of the boundary is affected by the bulk data, by its combinatorial structure and by the quantum correlations among the intertwiners
Summary
Entanglement is expected to play a crucial role in the emergence of spacetime and geometry from fundamental quantum entities, and to account for such emergence is a key goal of all quantum gravity approaches. Our work is a generalization of what has been done in the context of random tensor networks by Hayden et al in [31], with a crucial (with respect to the quantum gravity interpretation) change in perspective: the tensor networks we work with inherently possess a quantum-geometry characterization, being dual by construction to a triangulation of quantum space (as opposed to those used within the tensor networks/AdS correspondence [9], where such a characterization is implemented at a later stage, thanks to a definition of the metric in combinatorial terms) This implies that the corrections to the RyuTakayanagi entropy formula we find derive from the quantum-geometric properties of the bulk, and differ from that obtained in the AdS/CFT context (see, for example, [32]), which are related to the semiclassical. Focuses on a bulk reconstruction from the boundary density matrix, thereby following a different approach from the one used here, where properties of the bulk/boundary mapping are investigated through entanglement entropy evaluation
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