Bit threads and holographic entanglement

TL;DR

This work recasts holographic entanglement entropy through a max-flow/min-cut framework, introducing bit threads as divergenceless, norm-bounded flows whose maximum boundary flux equals the RT entropy, i.e. $S(A)=\max_v\int_A v$ with $|v|\le 1/(4G_{\rm N})$. It then demonstrates that information-theoretic quantities like $H(A|B)$, $I(A:B)$, and $I(A:B|C)$ acquire clear flow-based interpretations, yielding transparent proofs of subadditivity and SSA via the geometry of flows and their bottlenecks. The authors extend the picture to three regions, discuss the entanglement wedge interpretation, and explore connections to Weyl’s law as a possible bridge to quantum corrections and higher-derivative gravities. They also outline numerous open questions, including proving MMI within the flow language, extending to covariant and higher-derivative contexts, and using bit threads to illuminate emergent geometry from entanglement patterns. Overall, the bit-thread formalism offers a conceptually and computationally promising lens on holographic entanglement and spacetime emergence, linking information theory, geometry, and quantum gravity.

Abstract

The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, we invoke the notion of a "flow", defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness "bit threads". The entanglement entropy of a boundary region is given by the maximum flux out of it of any flow, or equivalently the maximum number of bit threads that can emanate from it. The threads thus represent entanglement between points on the boundary, and naturally implement the holographic principle. As we explain, this new picture clarifies several conceptual puzzles surrounding the RT formula. We give flow-based proofs of strong subadditivity and related properties; unlike the ones based on minimal surfaces, these proofs correspond in a transparent manner to the properties' information-theoretic meanings. We also briefly discuss certain technical advantages that the flows offer over minimal surfaces. In a mathematical appendix, we review the max flow-min cut theorem on networks and on Riemannian manifolds, and prove in the network case that the set of max flows varies Lipshitz continuously in the network parameters.

Figures (16)

• Figure 1: According to the Ryu-Takayanagi formula, \ref{['RT']}, the entanglement entropy $S(A)$ of a given boundary spatial region is given by the area of a corresponding bulk minimal surface $m(A)$.
• Figure 2: The minimal surface for the union of two separated regions undergoes a transition as a function of their separation between connecting them at small separation (left) and equalling the union of their respective minimal surfaces at large separation (right).
• Figure 3: Schematic illustration of the mutual information and conditional entropy in a classical system: Two correlated systems $A$ and $B$ can be encoded into $S(A)$ and $S(B)$ bits respectively, such that $I(A:B)$ bits of each are perfectly correlated, $H(A|B)$ bits of $A$ are uncorrelated with those of $B$, and $H(B|A)$ bits of $B$ are uncorrelated with those of $A$.
• Figure 4: The shaded boxes indicate pairs of bits that are maximally entangled between $A$ and $B$. Each such EPR pair contributes 1 to $S(A)$ and $S(B)$, 0 to $S(AB)$, 2 to $I(A:B)$, and $-1$ to $H(A|B)$. In this case, $H(A|B)$ is negative.
• Figure 5: Illustration of the Riemannian max-flow min-cut theorem: Given a boundary region $A$, the minimal-area representative of its homology class $m(A)$ is the bottleneck; its area gives an upper bound on the flux for any flow. The theorem asserts that there exists a flow $v(A)$ whose flux equals the area of $m(A)$ (times the constant $C$). In the figure, $v(A)$ is shown by its flow lines. On $m(A)$, this flow necessarily equals $C$ times the unit normal $n$.

Theorems & Definitions (4)

• Theorem A.1
• Lemma A.2
• proof
• Conjecture A.3