Covariant bit threads

TL;DR

This paper introduces new covariant reformulations of the holographic entanglement entropy formula, including minimax and flow-based approaches, which dynamically identify entanglement surfaces and wedges in a covariant setting.

Contribution

It presents novel covariant flow and minimax formulations of the HRT entanglement entropy, extending the bit thread concept to a covariant framework with convex programs.

Findings

Derived a minimax formula involving maximal achronal surfaces.

Formulated covariant V-flow and U-flow programs related by Lagrange duality.

Connected flow configurations to entanglement wedges and HRT surfaces.

Abstract

We derive several new reformulations of the Hubeny-Rangamani-Takayanagi covariant holographic entanglement entropy formula. These include: (1) a minimax formula, which involves finding a maximal-area achronal surface on a timelike hypersurface homologous to D(A) (the boundary causal domain of the region A whose entropy we are calculating) and minimizing over the hypersurface; (2) a max V-flow formula, in which we maximize the flux through D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm that is non-local in time; and (3) a min U-flow formula, in which we minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject to a lower bound on its norm that is non-local in space. The two flow formulas define convex programs and are related to each other by Lagrange duality. For each program, the optimal configurations dynamically find the HRT surface and the entanglement wedges of A and its complement. The V-flow formula is the covariant version of the Freedman-Headrick bit thread reformulation of the Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a "thread distribution", and explain how Riemannian flows, V-flows, and U-flows can be expressed in terms of thread distributions.