Soutenance de Qian Chen

General relativity is the most widely accepted gravitational theory to date which describes cosmological scale in terms of differential geometry of spacetime, and quantum theory is the most accurate physical theory which describes microscopic scale in terms of non-commutative algebra. While the two fundamental physical theories have not been put into one picture. Loop quantum gravity is a tentative theory of quantum gravity that inherits the essences from the two theories. One of the crucial properties of loop quantum gravity is diffeomorphism invariance, which, however, leads to the absence of locality in the theory. Inspired by the insights of holography and relational perspective, the issue involved to locality could be resolved by the prescriptions of quasi-locality and emergent geometry. Following the two insights, it is conceivable that the diffeomorphism invariant operators are entirely described by a theory living on boundary, and the quantum geometry could be reconstructed/emerged from entanglement and correlation. This thesis is dedicated to the relevant explorations: (1) to formulate a quasi-local description for loop quantum gravity with bulk-to-boundary relation; (2) to formulate quantum geometry in terms of the notions of quantum information theory. We wish these formulations shed light on the holography and renormalization flow in loop quantum gravity.

We study open spin networks, which are embedded in manifolds with non-empty boundaries, from the viewpoint of bulk-boundary maps. Based on it, we formulate quasi- local descriptions of loop quantum gravity. We investigate the coarse-graining procedure via tracing over bulk degrees of freedom, which encodes all that we can know about the quantum state of geometry from probing the boundary. We prove a boundary-to- bulk universal reconstruction procedure, to be understood as a purification of the mixed boundary state into a pure bulk state. We then move to define multipartite entanglement in spin networks and show the computation of entanglement excitation from holonomy operator, which also allows us to glimpse bulk curvature from entanglement. Moreover, by investigating another coarse-graining procedure - via gauge-fixing, which does not trace over any bulk degrees of freedom, we show a new interesting connection between bulk geometry and boundary observables via the dynamics of entanglement. Finally, we define the spin network entanglement between spin sub-networks, which correspond to spatial sub-regions. We then generalize the coarse-graining approach and prove that the entanglement between spin sub-networks is preserved under the coarse-graining (via gauge-fixing), which exhibits a holographic perspective for the topic of entanglement.