Abstract
In this thesis we explore the question of emergence of spacetime in quantum gravity. This question is particularly sharp in the classical limit of the AdS/CFT correspondence, where field theory states are mapped to Einstein-gravity spacetimes. We are interested in characterizing such states, for example, by studying necessary conditions for a state to have a geometric dual.
Holographic entanglement entropy (HEE) has proven to be a powerful tool for addressing these questions. Entanglement entropies of boundary regions on the field theory are computed holographically by the area of special extremal surfaces in spacetime. For static states, it is known that these areas, and thus the entropies, obey an infinite number of linear constraints known as holographic entropy inequalities (HEIs). Further, it is also known this problem can be mapped to the purely graph-theoretic one of computing min-cuts on weighted graphs. In other words, HEEs for static holographic states obey the same structural properties as min-cut functions.
The current state of affairs is somewhat awkward and mysterious. On the one hand, the full set of these constraints, their fundamental nature, and their physical interpretation is largely unknown. On the other hand, the mapping to a graph-theoretic framework raises questions about their physical significance: are these constraints fundamental, teaching us valuable physics, or are they merely artifacts of the non-generic nature of static states? We address both of these issues in this work.
In Chapter 2, we address the second point by providing strong numerical and analytical evidence that all known HEIs continue to hold for time-dependent states, focusing in particular on (2+1)-dimensional spacetimes.
In Chapter 3, we investigate the question of whether a graph theoretic construction exists in the fully covariant setting. We study a new formulation of covariant HEE, known as minimax, and formally prove many of its properties. The minimax formulation suggests a sufficient condition for the existence of a graph theoretic framework, which we provide evidence for and against.
In Chapter 4 we provide the strongest evidence to date that all HEIs hold covariantly in any spacetime dimension, by examining configurations in which violations, if they exist, are most likely to occur. In doing so, we observe a new combinatorial characterization of these inequalities which we formally study in Chapter 5 borrowing tools from majorization theory. We provide a series of combinatorial results that constraints the structure of the HEIs in the form of nested relational conditions between terms on the left-hand and right-hand sides.
All together, these results provide strong evidence that HEIs are genuine constraints on the entanglement structure of classical holographic states and that they contain dynamical information about the physics of holography.Further, these results initiate a new research program aimed at extracting physical insight from the HEIs using techniques from majorization theory and combinatorics.