Quantum Bit Threads and the Entropohedron

TL;DR

This work extends holographic entanglement entropy through quantum bit thread formalisms, introducing strict and cutoff-independent flow prescriptions that reproduce the QES formula for static states and remain robust in island/baby-universe settings. It unifies these flow-based approaches with a novel entanglement-structure object—the entanglement distribution function (EDF)—and its convex-geometric counterpart, the entropohedron, to encode how entanglement is distributed across boundary regions. The authors also develop quantum thread distributions and entanglement pair functions, providing multiple representations (flows, thread distributions, and graphs) that are equivalent under broad conditions, and they prove equivalence results with the QES prescription via convex duality. The framework offers regulator-independent insights, clarifies island behavior, and yields a structured way to quantify multipartite entanglement beyond area laws, with potential covariant extensions in future work. These tools deepen the connection between bulk geometry and boundary entanglement, offering new avenues to study holography, islands, and entanglement distributions in quantum gravity.

Abstract

We derive several new quantum bit thread prescriptions for holographic entanglement entropy, equivalent for static states to the quantum extremal surface formula. Our new prescriptions come in many varieties: vector field-based or based on measures over bulk curves, dependent or independent of the bulk UV regulator, loose and strict versions of constraints, and more. We also explore how bit threads behave in the presence of entanglement islands and baby universes. Finally, our prescriptions inspire new measures of entanglement that we call entanglement distribution functions, which can be packaged into a convex polytope that we call the entropohedron.

Figures (12)

• Figure 1: Contrasting the surface-based QES formula and a quantum bit thread prescription in an island setup. Both disks represent the same geometry, a time slice of an asymptotically AdS spacetime, and the same highly entangled bulk state. Left: the QES formula calculates $S(A)$ by minimising over bulk surfaces, $S(A) = \min_{\eth r \sim A } S_{\rm gen} (r) = S_{\rm gen} (a \cup I)$. Right: an optimal flow configuration for the quantum bit thread prescription \ref{['qmaxflow1']}. We maximise the boundary flux, $S(A) = \max_v \int_A n\cdot v$, with the $v$-constraints given in \ref{['qmaxflow1']}. The blue curves are the bit threads, which in this prescription are the integral curves of $v$, and the threads are maximally packed on $\eth a \cup \eth I$; the boundary of the island is a bit thread bottleneck.
• Figure 2: Quantum bit threads and closed universes I: the AdS bulk timeslice, the hyperbolic disk, is entangled with a closed universe $S^d$. The constraint in \ref{['qmaxflow2']} allows threads to end in the AdS bulk, but then also forces them to reappear in the closed universe, and then return to the AdS bulk, because it requires $\int_{\Sigma} \nabla \cdot v = 0$ for a pure boundary state, and we have taken $A = \partial \Sigma$. As depicted, the net maximal boundary flux is $\int_A n\cdot v = S(A) = 0$.
• Figure 3: Quantum bit threads and closed universes II: the left AdS bulk is entangled with the southern hemisphere of the $S^d$, and the northern hemisphere is entangled with the right AdS bulk. The constraint in \ref{['qmaxflow2']} allows threads to end in the left bulk, but they have to reappear in the sphere's southern hemisphere. They can then end in the northern hemisphere, but they have to reappear in the right bulk. In this example, as depicted, quantum bit threads in optimal configurations have to pass through the closed universe when going from $A$ to $A^c$.
• Figure 4: A doubly holographic setup. This figure illustrates the notation for geometric regions used in section \ref{['sec:flowsdouble']}. The entropy of region $r\subseteq \Sigma$ is proportional to the area of the RT surface $\eth \tilde{r}$ in the higher-dimensional second bulk $\tilde{\Sigma}$.
• Figure 5: Quantum bit threads in double holography. Left: a maximal flow configuration from the highest-dimensional bulk perspective. The bit threads are classical (divergenceless) and their field-lines move between the two bulks $\Sigma$ and $\tilde{\Sigma}$. $\eth \tilde{r}_{\text{QES}}$, the classical RT surface, is the bottleneck to the double flow $(v,\tilde{v})$. Right: the same flow configuration from the intermediate bulk perspective. Flow lines for $(v,\tilde{v})$ that are continuous from the perspective of $\tilde{\Sigma}$ become discontinuous flow lines for $v$ from the perspective of $\Sigma$.

Theorems & Definitions (57)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• Theorem 2.4