Geometric Aspects of Holographic Bit Threads

TL;DR

This work advances the geometric bit-thread picture of holographic entanglement entropy by providing explicit constructions of divergenceless vector fields that saturate the entanglement-bound and encode minimal-surface data. It introduces a general algorithm for building symmetric flows when the minimal surface is known, and a nesting-based approach to create maximally packed flows that interpolate between multiple nested regions, enabling multi-patch configurations. The authors apply these flows to disentangle quantum and thermal contributions via entanglement of purification and to demonstrate monogamy of mutual information through explicit multiflow realizations, including two disjoint intervals and BTZ setups. They also derive curvature- and energy-density-based constraints that ensure the flows remain valid under general geometric conditions, and discuss extensions, covariant formulations, and potential links to tensor-network pictures. Overall, the paper provides concrete, geometry-driven tools to study entanglement structures holographically, with broad implications for bulk reconstruction and quantum information in AdS/CFT.

Abstract

We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions.

Figures (12)

• Figure 1: Schematic representation of the conservation of flux, for an infinitesimal tube made of integral curves around a point $x_m$. The construction is local, so the $|V|$ obtained with this method is valid for all points away from the minimal surface. The final non-trivial check is to verify that the bound $|V|\leq1$ is respected everywhere.
• Figure 2: Vector lines and magnitude $|V|$ for a sphere in $d=1$ (orange), $d=2$ (blue) and $d=3$ (green) spatial dimensions, respectively. The vector field $V$ exhibits spherical symmetry so, for simplicity, we have plotted only one of the spatial axis in all case. The solid red line corresponds to the minimal surface, $m(A)$. This curve also signals the location where the magnitude of the vector field attains its maximal value, $|V|=1$.
• Figure 3: Vector lines for a strip in $d=2$ spatial dimensions (to be compared with the $d=1$ case, shown in Figure \ref{['Vspheres']}) and magnitude $|V|$ for $d=1$ (orange), $d=2$ (blue) and $d=3$ (green), respectively. The vector fields $V$ exhibit translational invariance along the transverse directions, which have been omitted for simplicity. The solid line(s) corresponds to the minimal surface(s), $m(A)$. These curves also signal the location where the magnitude of the vector fields attains its maximal value, $|V|=1$. In general, we observe that the integral curves get elongated and reach deeper into the bulk as the number of dimensions is increased. This property is inherited from their corresponding minimal surfaces.
• Figure 4: Typical integral curves and magnitude $|V|$ in a BTZ black hole geometry. The solid lines in blue correspond to the minimal surface, $m(A)$, and precisely at this location the magnitude of the vector field attains its maximal value, $|V|=1$. The vector lines in the shaded region correspond to threads that end at the horizon, while the the ones in the white area correspond to threads that go back to the boundary.
• Figure 5: An explicit construction of a maximally packed flow. In this example we have considered a family of intervals $A_n$ with left boundary located at $x_L=0$ and right boundary at $x_R=l_n$, with $l_{n+1}>l_n$. The two limiting minimal surfaces $m(A_0)$ and $m(A_N)$ bound the portion of the bulk that is shaded in green. In this region, the vector $V$ has maximal norm, i.e., $|V|=1$ and is orthogonal to the intermediate minimal surfaces $m(A_n)$. The UV cutoff that leaves the flux across the different surfaces constant is shown in red, but other choices are also allowed. The regions inside of $m(A_0)$ and outside of $m(A_N)$, which are shaded in blue, are continued with the geodesic flows constructed with the algorithm of section \ref{['section2']}.