Geometrizing the Partial Entanglement Entropy: from PEE Threads to Bit Threads

TL;DR

This paper introduces a geometric scheme to represent partial entanglement entropy in holographic CFTs using PEE threads, which are linked to bulk geodesics, and connects them to bit threads, providing a new perspective on entanglement structure in AdS/CFT.

Contribution

It develops a novel geometric framework for PEE using geodesic-based threads and relates these to bit threads, enhancing understanding of entanglement in holography.

Findings

PEE threads are represented by bulk geodesics connecting points.

A unique bit thread configuration can be derived from PEE threads for static regions.

RT formula can be reformulated as a minimization over weighted PEE threads.

Abstract

We give a scheme to geometrize the partial entanglement entropy (PEE) for holographic CFT in the context of AdS/CFT. More explicitly, given a point $\textbf{x}$ we geometrize the two-point PEEs between $\textbf{x}$ and any other points in terms of the bulk geodesics connecting these two points. We refer to these geodesics as the \textit{PEE threads}, which can be naturally regarded as the integral curves of a divergenceless vector field $V_{\textbf{x}}^{\mu}$, which we call \emph{PEE thread flow}. The norm of $V_{\textbf{x}}^{\mu}$ that characterizes the density of the PEE threads can be determined by some physical requirements of the PEE. We show that, for any static interval or spherical region $A$, a unique bit thread configuration can be generated from the PEE thread configuration determined by the state. Hence, the non-intrinsic bit threads are emergent from the intrinsic PEE threads. For static disconnected intervals, the vector fields describing a divergenceless flow is are longer suitable to reproduce the RT formula. We weight the PEE threads with the number of times it intersects with any homologous surface. Instead the RT formula is perfectly reformulated to be the minimization of the summation of the PEE threads with all possible assignment of weights.