Entanglement Threads in Holography

• Entanglement threads are one-dimensional geometric and information-theoretic constructs that represent quantum correlations within holographic duality frameworks like AdS/CFT.
• They utilize the max flow–min cut duality, bit threads, and partial entanglement entropy threads to operationalize entanglement measures and map boundary data to bulk geometries.
• The paradigm extends to multipartite systems, time-dependent settings, and island phases, providing insights into tensor networks, wormholes, and the reconstruction of emergent spacetime.

Entanglement threads are a family of geometric and information-theoretic constructs developed to represent and quantify the fine-grained structure of quantum correlations in holographic duality, notably within the AdS/CFT correspondence and its generalizations. In this framework, both classical and quantum entanglement in a boundary quantum field theory is encoded as a network of one-dimensional objects––threads––that traverse the bulk spacetime, offering both a precise operational reformulation of the Ryu–Takayanagi (RT) and quantum extremal surface (QES) prescriptions for entanglement entropy and an intuitive basis for reconstructing the emergent bulk geometry from entanglement data. A prominent innovation in recent years is the extension of this paradigm to arbitrary gravitational spacetimes, multipartite and partial entanglement structures, time-dependent (covariant) regimes, island phases, and holographic systems with nontrivial topologies such as wormholes.

1. Fundamental Principles and Mathematical Formulation

The modern entanglement thread paradigm synthesizes several core ideas (Freedman et al., 2016, Lin, 18 Jan 2025, Lin et al., 15 Jan 2024, Du et al., 6 Jun 2024):

• Max flow–min cut duality: The entanglement entropy $S(A)$ of a boundary region $A$ is equivalent to the maximum flux of a divergenceless, norm-bounded vector field $v$ ("flow") through $A$,

$$S(A) = \max_{v\,:\,\nabla\cdot v=0,\,|v| \leq 1/(4G_N)} \int_A v,$$

where $|v|$ is bounded pointwise, and each thread can be viewed as a "bit" of entanglement passing through the corresponding cross-sectional "bottleneck."

• Bit threads and entanglement threads: The integral curves of $v$ are termed "bit threads" and generalize the notion of entanglement threads: they encode not only bipartite entanglement between complementary regions but, with additional prescriptions, can capture partial, quantum, and multipartite entanglement as well (Harper, 2021, Lin, 18 Jan 2025).
• Partial Entanglement Entropy (PEE) threads: PEE threads represent the bulk "lift" of a two-point partial entanglement entropy function $\mathcal{I}(\vec{x},\vec{y})$ defined on the boundary, assigning to each pair $(\vec{x},\vec{y})$ a geodesic in the bulk. The density of such threads passing through any bulk co-dimension-2 surface $\Sigma$ is universal and given by $1/(4G)$ in pure AdS, so that

$$\mathcal{N}(\Sigma) = \frac{\operatorname{Area}(\Sigma)}{4G}$$

(Lin et al., 15 Jan 2024).

• Quantum bit threads: Quantum corrections (such as those arising from the Faulkner–Lewkowycz–Maldacena formula and generalized entropy) are incorporated by allowing $v$ to have divergence in the bulk according to a local entanglement density $s(x)$:

$\nabla \cdot v = -4G_N s(x)$

(Agón et al., 2021, Du et al., 6 Jun 2024).

• Covariant and dynamic generalizations: In time-dependent or Lorentzian contexts, dual "flow" programs have been formulated (max V-flow, min U-flow) with nonlocal bounds, preserving causality and allowing bit threads (now "V-threads" or "U-threads") to traverse both space and time, dynamically selecting the Hubeny–Rangamani–Takayanagi (HRT) surface (Headrick et al., 2022, Agón et al., 2019).

2. Geometric and Operational Construction

• Geodesics and kinematic space: Entanglement threads follow bulk geodesics whose endpoints are anchored to specified regions on the AdS boundary or a cutoff surface (Lin, 18 Jan 2025, Lin et al., 2023). Kinematic space—parametrizing all oriented boundary-anchored geodesics—provides a natural language for organizing threads and calculating derived quantities such as conditional mutual information and Crofton volume forms (Kudler-Flam et al., 2019).
• Thread flux and entropy relations: The number of threads crossing a minimal ("RT") surface $\gamma_A$ separating $A$ and its complement computes the entanglement entropy:

$S(A) = \frac{\operatorname{Area}(\gamma_A)}{4G_N}.$

This counting is realized both in the bit thread/max-flow prescription and, for PEE threads, by the intersection number with the PEE thread network (Lin et al., 15 Jan 2024).

• Rules for thread trajectories: In the most refined models, threads must:
  • Connect two basic boundary or cutoff regions, ensuring a "divergence-free" global organization;
  • Cross each relevant RT (or QES, in quantum/corrected prescriptions) surface exactly once;
  • Not cross additional RT surfaces more than once, to avoid overcounting entanglement (a direct reflection of properties such as strong subadditivity and monogamy) (Lin, 18 Jan 2025, Cui et al., 2018).

3. Extensions: Partial, Quantum, Covariant, and Island Phases

Partial Entanglement Entropy and Balanced Partial Entropy

• PEE threads allow for a fine-grained decomposition of EE into pairwise "PEE atoms" $\mathcal{I}(\vec{x},\vec{y})$, which are mapped into a continuous bulk network of geodesics. For any surface $\Sigma_A$ homologous to $A$, its intersection count with the PEE network yields the entropy, with the RT surface minimizing this count (Lin et al., 15 Jan 2024, Lin et al., 2023).
• Balanced Partial Entanglement Entropy (BPE) is a prescription to compute entanglement wedge cross-sections in complicated phases (such as the island phase) by appropriately combining PEE threads with multi-point data, crucially when surfaces must anchor on cutoff spheres due to the presence of a boundary or an "island" (Wen et al., 24 Aug 2024).

Quantum/Islands/General Spacetimes

• Quantum bit threads accommodate bulk entropy corrections by permitting threads to start or end in the bulk ("quantum threads") with divergence controlled by the bulk entanglement density $s(x)$. The resulting max-flow prescription for the generalized entropy recovers the QES/island formulas and encodes constraints such as lower bounds on bulk entanglement in the island region (Agón et al., 2021, Du et al., 6 Jun 2024).
• Islands and BCFT/braneworld extensions: In island phases, homologous surfaces are required to anchor on finite cutoff surfaces ("cutoff spheres") instead of the conformal boundary. The PEE thread distribution remains unchanged globally; only the anchoring prescription is altered, fully reproducing the island formula and correctly assigning entanglement wedge cross-sections (Wen et al., 24 Aug 2024).

Covariant and Time-Dependent Generalizations

• Covariant bit threads reformulate entanglement flows in Lorentzian spacetimes by maximizing (or minimizing) fluxes across causally defined regions, dynamically selecting the HRT surface, and incorporating causality via nonlocal, path-dependent norm bounds (Headrick et al., 2022, Agón et al., 2019). This formalism connects to the membrane theory of entanglement dynamics and supports derivations of entropy inequalities directly from flow nesting properties.

4. Multipartite and Hyperthread Structures

• Hyperthreads: A natural generalization to multipartite entanglement is achieved by "hyperthreads," or $k$-threads, which connect $k$ distinct boundary regions. The maximal number of $k$-threads is governed by the area of a special barrier surface $m_k$ via:

$$HP(\mathcal{A})_k = \frac{k}{\alpha} \operatorname{Area}(m_k)$$

where $\alpha$ counts intersections. This provides a geometric measure of genuine multipartite entanglement, extending the 2-thread (bit thread) case (Harper, 2021).

• Multicommodity flows and monogamy: Bit threads can be organized into families connecting all pairs of regions, and convex optimization approaches prove that the monogamy of mutual information (MMI) is built into the feasible thread (or multiflow) configurations (Cui et al., 2018, Headrick et al., 2020). For overlapping ("crossing") regions, relaxing local parallelism constraints can allow more general locking and flow partitionings, illuminating the structure of entropy inequalities and the limitations of locking in continuum versus network settings.

5. Applications to Tensor Networks, Thermal Geometry, and Holographic Reconstruction

• Tensor network realization: The refined thread prescription naturally translates to holographic tensor networks (HaPPY, hyperinvariant networks). Each thread becomes a "wire" of the network, perfectly matching RT area constraints and imparting a coarse-grained quantum circuit interpretation (Lin, 18 Jan 2025, Lin et al., 23 Aug 2025).
• BTZ and wormhole scenarios: In the case of the planar BTZ black hole and other wormhole geometries, entanglement threads traversing the wormhole horizon represent thermal (inter-boundary) entanglement, with explicit and regularized expressions for their fluxes. These flows support the interpretation of wormhole entropy contributions as due to perfect tensor-like entanglement (Lin et al., 23 Aug 2025).
• Bulk geometric reconstruction: The thread/PEE network approach allows for the Crofton formula to be realized operationally: any bulk area is reconstructible by counting thread intersections, explicitly relating "boundary" entanglement data to bulk geometry. This offers a bridge to integral geometry and new perspectives on the emergence of spacetime from entanglement structure (Lin et al., 15 Jan 2024, Lin et al., 2023).

6. Limitations, Open Problems, and Future Directions

• Nonuniqueness and convexity: While the bit thread prescription yields the correct entanglement entropy via max-flow/min-cut duality, individual thread configurations are highly nonunique. Entanglement threads as uniquely defined objects arise under the additional prescription that each contribution is fixed by conditional mutual information between minimally separated microscopic regions (Lin, 18 Jan 2025, Lin et al., 23 Aug 2025).
• Failure of nesting/geodesicity: For strip regions in $d > 2$, geodesic bit threads sometimes cannot be constructed due to the failure of the nesting property or nonintersecting geodesic assignment (Caggioli et al., 6 Mar 2024), and alternative constructions (e.g., minimal hypersurface-inspired flows) may be required.
• Quantum corrections and islands: The generalization to arbitrary gravitational spacetimes demonstrates that the bit thread (and entanglement thread) paradigm is robust, incorporating both geometric and quantum (bulk matter) corrections and setting fundamental bounds on the structure and distribution of quantum entanglement—especially in non-AdS, time-dependent, or island phases (Du et al., 6 Jun 2024, Wen et al., 24 Aug 2024).
• Future research: Directions include covariance and quantum extensions, multipartite capacities, connections to quantum error-correcting codes, tensor network fine-tuning, and integration with boundary conformal field theory or nonlocal modular Hamiltonians.

7. Connections to Broader Quantum Information and Holography

The entanglement thread formalism unites geometric, quantum-information, and tensor network perspectives, providing a physically transparent, mathematically precise, and computationally tractable set of tools for analyzing entanglement in quantum gravity and holography. Its integration of bit threads, PEE threads, hyperthreads, and their quantum generalizations establishes a rich universal language for the paper of both classical and quantum correlations in the emergent geometry of spacetime.