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Abstract

We introduce the geometric formulation of Quantum Mechanics in the quantum gravity context, and we use it to give a tensorial characterization of entanglement on spin network states. Starting from the simplest case of a single-link graph (Wilson line), we define a dictionary to construct a Riemannian metric tensor and a symplectic structure on the space of spin network states, showing how they fully encode the information about separability and entanglement, and, in particular, an entanglement monotone interpreted as a distance with respect to the separable state. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We extend then such analysis to the study of non-local correlations between two non-adjacent regions of a generic spin network. In the end, our analysis shows that the same spin network graph can be understood as an information graph whose connectivity encodes, both at the local and non-local level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.