Towards a Bit Threads Derivation of Holographic Entanglement of Purification

TL;DR

This work leverages the bit-thread formalism to relate holographic entanglement of purification to the entanglement wedge cross section. By constructing a geometric purification and a flux-maximizing thread configuration, the authors derive both an upper bound $E_P\le E_W$ and, under a carefully argued factorization and superposition framework, a matching lower bound, culminating in $E_P=E_W$ for holographic states under specific assumptions. The approach ties CMIs to thread counts and explores the role of purifying degrees of freedom encoded by bit threads, while also discussing minimal and geometric purifications and implications for black hole geometries. The results illuminate how bulk geometric data constrains boundary entanglement structure and suggest broader applicability to multipartite entanglement and holographic purification concepts.

Abstract

We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured $E_P = E_W$ relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.

Figures (7)

• Figure 1: Entanglement wedge for $AB$, bounded by the RT surfaces $\gamma_{AB}$. The entanglement wedge cross section $\Gamma$ is illustrated by the red dashed line.
• Figure 2: Types of bit threads that can cross the RT surface, $\gamma_{AB}$, of $AB$ for an arbitrary flow. The properties that distinguish different types of bit threads are where they begin and end ($A$, $B$, or $(AB)^c$), as well as how many times they intersect $\gamma_{AB}$ and the entanglement wedge cross section $\Gamma$ (red dashed line).
• Figure 3: Boundary subregions whose reduced density matrix approximately factorizes.
• Figure 4: Starting with a thread configuration that saturates the number of threads intersecting $\gamma_{AB}$, $\gamma_A$, and $\gamma_B$, as shown in (a) and described in Ref. Agon:2018lwq, the goal is to construct a configuration like (c), which also saturates $\Gamma$. (The parts of threads that cross the exterior component of $\gamma_{AB}$ and that lie outside the entanglement wedge have been suppressed in these diagrams.) Diagram (b) illustrates how this can be done by cutting and gluing threads. First, with a UV cutoff in place, it is helpful to think of there being a finite number of sites (yellow dots) on both $\gamma_{AB}$ and $\Gamma$ that threads must intersect. Initially, only $I(A:B)/2$ sites on $\Gamma$ are filled by threads, which form a tube running through $\Gamma$. Divide the remaining $E_W(A:B) - I(A:B)/2$ sites into a group of $n_{\mathrm{hi}}$ sites above the tube and $n_{\mathrm{lo}}$ sites below the tube. (In two spatial dimensions the division is unique, but in higher dimensions there may be freedom in this choice.) Then, cut the $I(A:B)/2$ threads that cross $\Gamma$, as well as $n_{\mathrm{hi}}$ threads that intersect the exterior component of $\gamma_{AB}$ on the side adjacent to $B$ and $n_{\mathrm{lo}}$ threads that intersect the interior component of $\gamma_{AB}$ on the side adjacent to $A$. The locations of the cuts are indicated by red crosses in (a). Finally, glue adjacent cut threads together as shown in (b) to arrive at the maximizing configuration (c). This cutting-and-gluing procedure is equivalent to "combing" the original thread configuration, as depicted in (d). Combing means dragging $n_{\mathrm{hi}}$ threads that intersect sites on the $A$-adjacent side of the exterior component of $\gamma_{AB}$ over to sites on the $B$-adjacent side, and vice-versa for $n_{\mathrm{lo}}$ sites on the interior component of $\gamma_{AB}$. The only additional subtlety is that, with a UV cutoff in place, we must think of $n_{\mathrm{hi}}$ threads being nucleated from the UV where the exterior component of $\gamma_{AB}$ meets $A$ and $n_{\mathrm{hi}}$ sites being absorbed by the UV where the exterior component of $\gamma_{AB}$ meets $B$. The same is true for $n_{\mathrm{lo}}$ threads hitting the interior component, with $A$ and $B$ flipped. Although the diagrams as we have drawn them here are directly reflective of AdS3/CFT2, we do not believe that there are any barriers to combing and cutting-and-gluing in arbitrary dimensions.
• Figure 5: Representative examples of bit thread configurations simultaneously maximizing the flow through the entanglement wedge surface $\Gamma$ (red dashed line) and RT surface $\gamma_{AB}$ (green line). The vector fields specifying the flow are indicated on $\gamma_{AB}$ and $\Gamma$ by arrows, and the individual RT surfaces for $A$ and $B$ are depicted with purple dashed lines. Left: $A$ and $B$ are two surfaces defining a proper subregion of a single boundary, on which the CFT state is pure. Right: $A$ and $B$ partition an entire boundary, on which is defined a mixed CFT state, resulting in a horizon in the bulk.