Partial Entanglement Entropy Threads

• Partial Entanglement Entropy (PEE) threads are a continuum network of weighted bulk geodesics that decompose von Neumann entropy via a unique bi-local kernel.
• The formalism employs the Crofton formula and the RT prescription, ensuring additivity, permutation symmetry, and invariance under local unitaries.
• This approach refines the bit thread model by offering detailed insights into entanglement contours, tensor networks, and quantum corrections in holography.

Partial Entanglement Entropy (PEE) threads are a geometric and information-theoretic refinement of the holographic entanglement structure in AdS/CFT, expressing the fine-grained contributions to von Neumann entropy via a bi-local density associated to infinitesimal region pairs. The PEE thread formalism encodes entanglement entropy as a continuum network of bulk geodesics—“threads”—with density precisely fixed by the boundary PEE structure. This construction not only provides a unique decomposition of entanglement entropy, compatible with physical and symmetry constraints, but also establishes an exact correspondence between the Crofton formula in integral geometry and the Ryu–Takayanagi (RT) prescription for holographic entanglement entropy. PEE threads generalize and refine the bit-thread picture, are intimately connected to entanglement contours, and play a central role in understanding tensor network models, quantum corrections, and information flow in holographic phases including those with islands.

1. Mathematical Formulation and Physical Requirements

The PEE $s_{\mathcal{A}}(\mathcal{A}_i)$ quantifies the contribution of a subset $\mathcal{A}_i \subset \mathcal{A}$ to the entropy $S_\mathcal{A}$. Any viable PEE satisfies:

• Additivity: $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$ for disjoint $\mathcal{A}_i, \mathcal{A}_j$.
• Permutation symmetry: $$s_\mathcal{A}(\mathcal{A}_i) = s_{\bar{\mathcal{A}}}(\bar{\mathcal{A}}_i)$$, with $\bar{\mathcal{A}}_i$ the complement in the purification.
• Normalization: $s_\mathcal{A}(\mathcal{A}) = S_\mathcal{A}$.
• Positivity and upper bound: $$0 \le s_\mathcal{A}(\mathcal{A}_i) \le S_{\mathcal{A}_i}$$.
• Invariance under local unitaries, and under global symmetries.

These requirements single out a unique Poincaré-invariant, cutoff-independent bi-local kernel $\mathcal{I}(x, y)$ such that the mutual PEE between infinitesimal regions at $\mathcal{A}_i \subset \mathcal{A}$0 and $\mathcal{A}_i \subset \mathcal{A}$1 is

$\mathcal{A}_i \subset \mathcal{A}$2

where $\mathcal{A}_i \subset \mathcal{A}$3 is the central charge and $\mathcal{A}_i \subset \mathcal{A}$4 is the area of the unit $\mathcal{A}_i \subset \mathcal{A}$5-sphere (Wen, 2019, Lin et al., 2024). The total entropy of a region $\mathcal{A}_i \subset \mathcal{A}$6 is expressed as

$\mathcal{A}_i \subset \mathcal{A}$7

subject to an appropriate regulator. For subregion $\mathcal{A}_i \subset \mathcal{A}$8, the PEE is $\mathcal{A}_i \subset \mathcal{A}$9 (Lin et al., 2024, Wen et al., 22 Dec 2025).

2. PEE Threads: Bulk Geodesic Web and Density Theorems

The "PEE thread" associated to points $S_\mathcal{A}$0 is the unique spacelike bulk geodesic $S_\mathcal{A}$1 connecting $S_\mathcal{A}$2 and $S_\mathcal{A}$3. The continuous set of all such geodesics, each equipped with weight $S_\mathcal{A}$4, forms a web—termed the "PEE network"—that tessellates the AdS bulk (Wen et al., 22 Dec 2025, Lin et al., 2024). The network obeys the following fundamental density law: $S_\mathcal{A}$5 uniformly throughout the bulk. Thus, the number of thread intersections with any surface is proportional to its area, leading directly to

$S_\mathcal{A}$6

for any (not necessarily extremal) hypersurface $S_\mathcal{A}$7. This law underpins the translation between integral geometry (Crofton formula) and holographic entropy (Lin et al., 2024, Wen et al., 22 Dec 2025).

3. RT Formula and Weighted Thread Reformulation

For a static boundary region $S_\mathcal{A}$8, the minimal intersection number of the PEE network with any homologous bulk surface is attained by the RT surface $S_\mathcal{A}$9: $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$0 For connected regions, this reduces to the flux through $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$1 of the superposed vector flow $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$2. For disconnected regions, each thread is counted with its intersection number (“weight”) $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$3 with $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$4, and the entropy is the minimum weighted sum over all homologous surfaces (Lin et al., 2023, Lin et al., 2024).

4. Quantum Corrections, Modular Hamiltonian, and Contour First Law

PEE threads provide a fine-grained perspective on quantum corrections to entanglement entropy. At leading order in $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$5, the bulk contribution $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$6 to $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$7 is encoded in the subset of threads that penetrate into the entanglement wedge and terminate in the interior:

• "Modular-slice" analysis shows the quantum correction to the boundary PEE aligns with the bulk PEE for the corresponding entanglement wedge region (Han et al., 2021).
• The first-law-like relation for the entanglement contour,

$$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$8

with $$s_{\mathcal{A}}(\mathcal{A}_i \cup \mathcal{A}_j) = s_{\mathcal{A}}(\mathcal{A}_i) + s_{\mathcal{A}}(\mathcal{A}_j)$$9 the modular-Hamiltonian contour, has a threads interpretation: perturbing the stress tensor deforms the local density of threads, making $\mathcal{A}_i, \mathcal{A}_j$0 the local “equation of motion” for the flow (Han et al., 2021).

5. Extensions: BCFTs, Island Phases, and Tensor Networks

In BCFTs and systems with entanglement islands, PEE threads generalize naturally:

• In island phases, boundary points are replaced by cutoff spheres, and the network of geodesics terminates on these surfaces rather than the asymptotic boundary, reproducing the island formula for entropy by minimization over such anchored surfaces (Wen et al., 2024).
• Balanced partial entanglement entropy (BPE), defined via flux balance conditions in the thread network, reproduces the entanglement wedge cross section (EWCS) even in the presence of islands (Wen et al., 2024, Lin et al., 2021).
• The PEE network can be quantized to yield tensor network models, including factorized networks (tensor products of EPR pairs along PEE threads) and random tensor networks. In both "factorized" and "random" PEE tensor networks, the entanglement entropy of any boundary region is determined precisely by the minimal number of thread cuts along a homologous bulk surface, reproducing the RT formula (Wen et al., 22 Dec 2025).

6. Relation to Bit Threads, Conditional Mutual Information, and Multipartite Structure

• PEE threads refine the bit thread formalism by uniquely encoding the conditional mutual information structure between infinitesimal regions and their complements. The thread density for a pair $\mathcal{A}_i, \mathcal{A}_j$1 is directly equal to $\mathcal{A}_i, \mathcal{A}_j$2, which, for suitable decompositions, aligns with conditional mutual information or multipartite extensions (hyperthreads/perfect tensors) for more complex regions (Lin, 2023, Lin et al., 23 Aug 2025).
• The thread-state correspondence further clarifies that each PEE thread corresponds to a Bell pair (2-thread) or, for higher multipartitions, to an absolutely maximally entangled perfect tensor state. For disconnected regions, correct entropic accounting requires this refinement beyond bipartite threads (Lin, 2023).

7. Integral Geometry and the Crofton Formula

• The Crofton formula in integral geometry states that the area of any bulk co-dimension one surface can be written as an average of intersection numbers over the space of geodesics:

$\mathcal{A}_i, \mathcal{A}_j$3

Substituting the density and identifying $\mathcal{A}_i, \mathcal{A}_j$4 with the bi-local PEE weight reproduces the PEE-thread intersection law (Wen et al., 22 Dec 2025, Lin et al., 2024).

• This correspondence establishes an explicit geometric encoding of boundary entanglement structure and provides a direct “weaving” of AdS geometry by entanglement threads.

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Key References:

• (Wen et al., 22 Dec 2025) "Holographic Tensor Networks as Tessellations of Geometry"
• (Lin et al., 2024) "Weaving the (AdS) spaces with partial entanglement entropy threads"
• (Lin et al., 2023) "Geometrizing the Partial Entanglement Entropy: from PEE Threads to Bit Threads"
• (Wen et al., 2024) "Partial entanglement entropy threads in island phase"
• (Han et al., 2021) "First Law and Quantum Correction for Holographic Entanglement Contour"
• (Wen, 2019) "Formulas for Partial Entanglement Entropy"
• (Lin, 2023) "Distilled density matrices of holographic PEE from thread-state correspondence"
• (Lin et al., 23 Aug 2025) "The holographic entanglement pattern of BTZ planar black hole from a thread perspective"