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arXiv:2608.19428

Holographic Bit Threads from String-Diagrammatic Quantum Information Flow

Yi-Yu Lin, Song Cheng

hep-thquant-phSubmitted Aug 19, 2026• Announced Aug 21, 2026• 0 citations• Revised Aug 26, 2026 (v2)
v1 · Aug 19, 2026v2 · Aug 26, 2026
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Abstract

Bit threads, arising as the convex dual of the minimal-surface formula for holographic entanglement entropy, are line-like structures with nontrivial bulk trajectories. Their physical picture has long been associated with Bell-pair interpretations, yet the meaning of their detailed trajectories, particularly their nonuniqueness, remains unclear. We propose to apply Coecke's notion of quantum information flow (QIF) in protocol-network geometry, developed within categorical quantum mechanics, to a holographic setup, and show that its trajectories within the discrete holographic bulk---the tensor network---obey the divergence-free and density-bound conditions of bit threads. From this process-centered perspective, bit-thread nonuniqueness becomes natural: for a given resource state, different protocols realizing the same distillation task can give rise to different QIF trajectories. We further analyze stabilizer-type holographic tensor networks using ZX string diagrams and show that QIF trajectories can probe finer entanglement structure beyond the level captured by entanglement entropy.

Comments
20 pages, 16 figures

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Bit threads, arising as the convex dual of the minimal-surface formula for holographic entanglement entropy, are line-like structures with nontrivial bulk trajectories. Their physical picture has long been associated with Bell-pair interpretations, yet the meaning of their detailed trajectories, particularly their nonuniqueness, remains unclear. We propose to apply Coecke's notion of quantum information flow (QIF) in protocol-network geometry, developed within categorical quantum mechanics, to a holographic setup, and show that its trajectories within the discrete holographic bulk---the tensor network---obey the divergence-free and density-bound conditions of bit threads. From this process-centered perspective, bit-thread nonuniqueness becomes natural: for a given resource state, different protocols realizing the same distillation task can give rise to different QIF trajectories. We further analyze stabilizer-type holographic tensor networks using ZX string diagrams and show that QIF trajectories can probe finer entanglement structure beyond the level captured by entanglement entropy.
A bstract 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.
Abstract Quantum corrections to holographic entanglement entropy require knowledge of the bulk quantum state. In this paper, we derive a novel dual prescription for the generalized entropy that allows us to interpret the leading quantum corrections in a geometric way with minimal input from the bulk state. The equivalence is proven using tools borrowed from convex optimization. The new prescription does not involve bulk surfaces but instead uses a generalized notion of a flow, which allows for possible sources or sinks in the bulk geometry. In its discrete version, our prescription can alternatively be interpreted in terms of a set of Planck-thickness bit threads, which can be either classical or quantum. This interpretation uncovers an aspect of the generalized entropy that admits a neat information-theoretic description, namely, the fact that the quantum corrections can be cast in terms of entanglement distillation of the bulk state. We also prove some general properties of our prescription, including nesting and a quantum version of the max multiflow theorem. These properties are used to verify that our proposal respects known inequalities that a von Neumann entropy must satisfy, including subadditivity and strong subadditivity, as well as to investigate the fate of the holographic monogamy. Finally, using the Iyer-Wald formalism we show that for cases with a local modular Hamiltonian there is always a canonical solution to the program that exploits the property of bulk locality. Combining with previous results by Swingle and Van Raamsdonk, we show that the con- sistency of this special solution requires the semi-classical Einstein’s equations to hold for any consistent perturbative bulk quantum state.
We investigate the relations between bit thread, entanglement distillation, and entanglement of purification in the holographic framework. Specifically, we give a bit thread interpretation for the one-shot entanglement distillation (OSED) tensor network, which can be understood as reconstructing the geometric structure of the bulk spacetime from the entanglement information of the boundary quantum system through the ``surface growth scheme.'' Moreover, by showing that the holographic entanglement of purification (EoP) process can be regarded as a special case of our ``surface growth scheme,'' we naturally obtain the bit thread interpretation of the holographic EoP in our framework, which turns out to be different from the existing interpretations. Since our interpretation for holographic EoP is obtained in a more general and physical framework, the advantage of our version is that it is more natural, and possibly more reasonable, to reflect the real physical entanglement structures.
Abstract Bit threads, a dual description of the Ryu-Takyanagi formula for holographic entanglement entropy (EE), can be interpreted as a distillation of the quantum information to a collection of Bell pairs between different boundary regions. In this article we discuss a generalization to hyperthreads which can connect more than two boundary regions leading to a rich and diverse class of convex programs. By modeling the contributions of different species of hyperthreads to the EEs of perfect tensors we argue that this framework may be useful for helping us to begin to probe the multipartite entanglement of holographic systems. Furthermore, we demonstrate how this technology can potentially be used to understand holographic entropy cone inequalities and may provide an avenue to address issues of locking.
Abstract We use the ‘bit thread’ formulation of holographic entanglement entropy to highlight the distinction between the universally-valid strong subadditivity and the more restrictive relation called monogamy of mutual information (MMI), known to hold for geometrical states (i.e. states of holographic theories with gravitational duals describing a classical bulk geometry). In particular, we provide a novel proof of MMI, using bit threads directly. To this end, we present an explicit geometrical construction of cooperative flows which we build out of disjoint thread bundles. We conjecture that our method applies in a wide class of configurations, including ones with non-trivial topology, causal structure, and time dependence. The explicit nature of the construction reveals that MMI is more deeply rooted in bulk locality than is the case for strong subadditivity.
Motivated by the ability to consistently apply the Ryu-Takayanagi prescription for general convex surfaces and the relationship between entanglement and geometry in tensor networks, we introduce a novel, covariant bulk object---the holographic slice. The holographic slice is found by considering the continual removal of short-range information in a boundary state. It thus provides a natural interpretation as the bulk dual of a series of coarse-grained holographic states. The slice possesses many desirable properties that provide consistency checks for its boundary interpretation. These include the monotonicity of both area and entanglement entropy, uniqueness, and the inability to probe beyond late-time black hole horizons. Additionally, the holographic slice illuminates physics behind entanglement shadows, as minimal-area extremal surfaces anchored to a coarse-grained boundary may probe entanglement shadows. This lets the slice flow through shadows. To aid in developing intuition for these slices, many explicit examples of holographic slices are investigated. Finally, the relationship to tensor networks and renormalization (particularly in AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT) is discussed.