Bit Threads: From Entanglement to Geometric Entropies

TL;DR

The paper develops a covariant phase space–based construction of bit threads in holography, showing how conserved CPS currents can generate divergenceless flows that encode holographic entropy. It demonstrates that CPS-derived flows reproduce the standard bit-thread results in cases where the RT surface coincides with a horizon and identifies a necessary harmonic gauge correction v_d in more general backgrounds; it then extends the flow picture to Wald entropy and differential entropy, reformulates the first laws of entropy in terms of bit-thread flows, and discusses quantum corrections via relative entropy and the dominant energy condition, as well as applications to time-dependent AdS-Vaidya spacetimes. The work provides a unified geometric framework linking entanglement, black hole thermodynamics, and bulk dynamics, with potential extensions to covariant threads and higher-derivative gravity. Overall, it offers practical methods for constructing thread configurations in dynamical spacetimes and for exploring the interplay between bulk energy conditions and boundary entanglement.

Abstract

In this work, we attempt to construct bit thread configurations for various backgrounds using expressions from the covariant phase space formalism. We find that when the Ryu-Takayanagi surface is same as the horizon, such expressions are sufficient. In other cases, it differs by gradient of a harmonic function. We explore its relation to Wald and differential entropy, and re-express the first law of entanglement entropy in terms of bit threads. Inclusion of quantum effects imposes some constraints on the bulk entanglement via the dominant energy condition. We also apply our method to ascertain a bit thread configuration in a certain dynamical spacetime.

Figures (3)

• Figure 1: Flow lines (black curves with arrows) for bit threads obtained from transforming a valid thread configuration in AdS-Rindler to Poincaré. The grey horizontal line denotes the boundary, and the red semicircle is the RT surface corresponding to the boundary subregion (shown in blue) $A\in [-2,2]$.
• Figure 2: Schematic diagram of a Riemannian manifold showing a flow $\upsilon(A)$ and a surface $m(A)$ with minimal area, homologous to the boundary region $A$. The Riemannian MFMC tells that the flux of the flow (flow lines shown in brown) is proportional to the area of the surface $m(A)$ (shown in blue). This minimal surface acts as the bottleneck for the flow.
• Figure 3: Flow lines for bit threads obtained from solving the PDE: $\nabla_a v_{PDE}^a=0$ and imposing the conditions $v_{PDE}^a|_{RT} =n^a$ and $|v_{PDE}| \le 1$. The blue horizontal line denotes the boundary subregion, and the red semicircle is the corresponding RT surface.