We study the relationship between the geometry of spacetime and quantum information. This is motivated by recent insights which suggest that geometry is an emergent phenomenon in quantum gravity, and in particular that geometry is built from quantum entanglement.
Part I of this thesis is focused on the relationship between area and entropy. Area/entropy relations are ubiquitous in gravitating systems. One manifestation of this relationship comes from the AdS/CFT correspondence, which posits a duality between quantum gravity in asymptotically Anti-de Sitter space and certain quantum field theories that can be thought of as living on the boundary of the Anti-de Sitter spacetime.
When the field theory has a large number of strongly coupled fields, the dual quantum gravity spacetime is described to good approximation by classical General Relativity. In this limit, the Hubeny-Rangamani-Takayanagi (HRT) formula relates the area of surfaces in the bulk spacetime to the entanglement entropy of associated subregions in the dual field theory.
A second potential incarnation of the relationship between area and entropy comes in the form of black hole area laws and a more locally-defined generalization known as a holographic screens. We explore connections between these different notions of area and entropy by studying the properties of holographic screens in Anti-de Sitter space. We also study a (modified version) of HRT like surfaces attached to arbitrary boundaries (that need not be an Anti-de Sitter boundary).
Part II of this thesis involves the study of traversable wormholes. Physicists have long believed that wormholes that could be crossed by an observer or signal would be impossible to build. In fact it can be shown that, with only classical matter, traversable wormholes cannot exist. While it remained possible that subtle quantum effects might be able to provide the negative energy needed to build them, there were no successful attempts at doing so. Recently, however, examples were constructed in AdS that relied on putting interactions in the dual, entangled quantum systems, and thus illustrated the intimate relation between quantum entanglement and spacetime geometry. Below, we describe a general method by which to construct traversable wormholes that can be applied to any spacetime, including asymptotically flat space. We explicitly construct several examples in AdS and in flat space, and generalize the result to construct multi-mouthed wormholes. We further use these multi-mouthed wormholes to study the entanglement structure of the spacetime they reside in.