Bit Threads and the Membrane Theory of Entanglement Dynamics

TL;DR

This paper reformulates the membrane theory of entanglement dynamics as a max flow problem using the bit threads approach, establishing a duality that connects hydrodynamic entanglement models with holographic surface extremization.

Contribution

It proves a max flow-min cut theorem for the membrane theory and links it explicitly to the holographic max flow program, advancing the understanding of entanglement in chaotic systems.

Findings

Membrane theory can be expressed as a max flow problem.

Established a duality between membrane theory and holographic extremization.

Derived the max flow program from the holographic prescription in the hydrodynamic regime.

Abstract

Recently, an effective {\it membrane theory} was proposed that describes the ``hydrodynamic'' regime of the entanglement dynamics for general chaotic systems. Motivated by the new {\it bit threads} formulation of holographic entanglement entropy, given in terms of a convex optimization problem based on flow maximization, or equivalently tight packing of bit threads, we reformulate the membrane theory as a max flow problem by proving a max flow-min cut theorem. In the context of holography, we explain the relation between the max flow program dual to the membrane theory and the max flow program dual to the holographic surface extremization prescription by providing an explicit map from the membrane to the bulk, and derive the former from the latter in the ``hydrodynamic'' regime without reference to minimal surfaces or membranes.