Bulk Entanglement Gravity and Radon Transform: Towards Finding Einstein's Equation in Hilbert Space

We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, such as ones generated by short-range entangled tensor networks, an emergent spatial geometry along with its best-fit dimensionality can be determined from an amorphous configuration. We show how Radon transforms can be used to convert quantum mutual information data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a spacetime geometry, and we argue that using a generalized version of Ryu-Takayanagi formula the geometry should obey Einstein's equation in the weak-field limit near Minkowski spacetime. The results of emergent geometry and gravity can also be applied to tensor networks. It suggests that more generic complex quantum systems, such as subsystem quantum error correction codes, can contain natural ingredients necessary for emergent gravity.