Abstract
This thesis is the beginning of an investigation into the multipartite entanglement of holographic systems. To accomplish this we make use of tools from convex optimization which allow for a recasting of the well known Ryu-Takayangi (RT) formula for holographic entanglement entropy. In this alternate formulation the entanglement can be thought of as a maximal configuration of bit threads which should be thought of as a distillation of the holographic state to Bell pairs. This technology can be applied to other aspects of holography beyond the calculation of simple RT surfaces. We consider two such applications:
In the first we examine the dual programs of another surface of interest: the entanglement wedge cross section. It is found that these dual programs introduce constraints on threads which isolate them to certain regions of the geometry. In the case of three or more boundary regions the multipartite entanglement wedge cross section, which is potentially related to multipartite entanglement, can be considered. This introduces non-trivial constraints between different “species” of threads. It is also shown that these programs can be uplifted to a manifold of nontrivial topology. The multipartite entanglement wedge cross section can be mapped to a minimal surface of a particular nontrivial homology class which is dual to a configuration of threads which cross different copies of the original geometry. This demonstrates a possible connection between topology and multipartite entanglement in holographic systems.
In the second we generalize bit threads to “k-hyperthreads” or just “k-threads” which connect to k distinct boundary regions. The idea being that these objects should in analogy with the usual 2-threads be identified with distillable units of entanglement. After developing the necessary technology by making use of convex duality we show that in particular hyperthreads modeled after perfect tensor states seem particularly well adapted to applications of holography. This leads to a conjecture that maximal configurations of these perfect tensor hyperthreads are capable of locking, or reproducing, the full entropy vector: the var- ious possible entanglement entropies one can consider for a particular choice of boundary regions. If so this leads to a natural conjecture about the distillation of holographic states to specific combinations of perfect tensor states which preserve the entropy vector.