Crossing versus locking: Bit threads and continuum multiflows

TL;DR

This paper explores how different density bounds affect the ability to lock boundary regions in holographic spacetimes using bit threads, revealing conditions under which regions can or cannot be simultaneously represented.

Contribution

It introduces a continuum multiflow framework for bit threads, analyzing how density bounds influence locking of boundary regions and establishing new conditions for lockability.

Findings

Under the most stringent bound, non-crossing regions can be locked.

Crossing regions cannot be locked under the most stringent bound.

A less stringent bound allows locking of crossing pairs.

Abstract

Bit threads are curves in holographic spacetimes that manifest boundary entanglement, and are represented mathematically by continuum analogues of network flows or multiflows. Subject to a density bound, the maximum number of threads connecting a boundary region to its complement computes the Ryu-Takayanagi entropy. When considering several regions at the same time, for example in proving entropy inequalities, there are various inequivalent density bounds that can be imposed. We investigate for which choices of bound a given set of boundary regions can be "locked", in other words can have their entropies computed by a single thread configuration. We show that under the most stringent bound, which requires the threads to be locally parallel, non-crossing regions can in general be locked, but crossing regions cannot, where two regions are said to cross if they partially overlap and do not cover the entire boundary. We also show that, under a certain less stringent density bound, a crossing pair can be locked, and conjecture that any set of regions not containing a pairwise crossing triple can be locked, analogously to the situation for networks.