UC Berkeley

There is a deep connection between entanglement and geometry in quantum gravity. One manifestation of this connection is through the generalized entropy: the sum of the area of Cauchy-dividing surfaces in Planck units and the von Neumann entropy of the fields on one side of the surface. The generalized entropy is expected to satisfy fundamental thermodynamical conditions which in certain limits result in quantum field theory (QFT) inequalities relating the energy density and the shape derivatives of the von Neumann entropy, most notably the quantum null energy condition (QNEC). This dissertation is devoted to studying various aspect of the generalized entropy, the QNEC, and other information-theoretic aspects of QFT. First, we will use the QNEC to demonstrate the locality of the logarithm of the vacuum density matrix in certain regions of any holographic conformal field theory (CFT). We will then holographically prove the QNEC in general curved spacetime of less than six spacetime dimensions. Next, we will demonstrate a certain limit of the QNEC where an equality between certain components of the stress tensor and the second shape derivative of von Neumann entropy emerges. We will then focus on the study of generalized entropy. Within the AdS/CFT framework, we find a microscopic description for the generalized entropy of black holes using a geometric construction. We then apply this construction to extremal surface in AdS, which constitutes the bulk dual of the Connes cocycle flow on the boundary CFT. Finally, we use propose a novel lower bound on the total energy of spacetime involving the generalized entropy on certain light-sheets.