Bit threads and holographic entanglement

TL;DR

This paper introduces a flow-based reformulation of the Ryu-Takayanagi formula for holographic entanglement entropy, using bit threads to provide clearer conceptual understanding and new proofs of key properties.

Contribution

It presents a novel flow-based approach to holographic entanglement entropy, replacing minimal surfaces with divergenceless vector fields called bit threads, and offers new proofs of entropy inequalities.

Findings

Flow-based proofs of strong subadditivity

Bit threads clarify entanglement structure

Technical advantages over minimal surface methods

Abstract

The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, we invoke the notion of a "flow", defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness "bit threads". The entanglement entropy of a boundary region is given by the maximum flux out of it of any flow, or equivalently the maximum number of bit threads that can emanate from it. The threads thus represent entanglement between points on the boundary, and naturally implement the holographic principle. As we explain, this new picture clarifies several conceptual puzzles surrounding the RT formula. We give flow-based proofs of strong subadditivity and related properties; unlike the ones based on minimal surfaces, these proofs correspond in a transparent manner to the properties' information-theoretic meanings. We also briefly discuss certain technical advantages that the flows offer over minimal surfaces. In a mathematical appendix, we review the max flow-min cut theorem on networks and on Riemannian manifolds, and prove in the network case that the set of max flows varies Lipshitz continuously in the network parameters.