Flux Function of Entanglement Threads in Holography

• Flux Function of Entanglement Threads is a quantitative method that decomposes total entanglement entropy into contributions from pairs of boundary subregions using conditional mutual information.
• It refines traditional bit thread models by uniquely attributing fluxes to both internal and horizon-crossing entanglement channels, mapping microscopic tensor network structures.
• The framework links partial entanglement entropy with perfect tensor properties, offering insights into holographic error correction in BTZ black holes.

The flux function of entanglement threads provides a fine-grained, quantitative decomposition of boundary entanglement entropy in holographic systems, particularly addressing the structure of quantum entanglement in the BTZ planar black hole background. Unlike classical bit thread formulations, which emphasize total flux and density bounds to capture entanglement entropy, the entanglement thread framework assigns unique, conditional mutual information–based fluxes between elementary subregions, yielding a detailed accounting of the sources contributing to the von Neumann entropy. This approach refines the tensor network analogy for holography and elucidates the emergence of perfect-type (absolutely maximally entangled) structures due to the interplay of internal and wormhole-traversing threads (Lin et al., 23 Aug 2025).

1. Flux Function Derivation

The flux function is derived by partitioning the spatial boundary into a set of elementary subregions $\{A_i\}$ and associating to each unordered pair $(A_i, A_j)$ a flux $F_{ij}$ that quantifies the number of entanglement threads connecting $A_i$ and $A_j$. Mathematically, this is defined via the conditional mutual information (CMI) as

$$F_{ij} \equiv \tfrac{1}{2} I(A_i, A_j \mid \widehat{A})$$

where $\widehat{A}$ denotes the (possibly empty) region separating $A_i$ and $A_j$ on the boundary.

For the pure $\text{AdS}_3$ geometry, on a Poincaré disk, the flux function evaluates to

$$F_{ij} = \frac{c}{6} \ln \left( \frac{\sin \frac{|\xi_{i_1} - \xi_{j_1}|}{2} \sin \frac{|\xi_{i_2} - \xi_{j_2}|}{2}}{ \sin \frac{|\xi_{i_1} - \xi_{j_2}|}{2} \sin \frac{|\xi_{i_2} - \xi_{j_1}|}{2} } \right)$$

where $\xi_{i_1}, \xi_{i_2}$ are the endpoint coordinates of $A_i$; analogous expressions with trigonometric functions replaced by hyperbolic sines hold in the BTZ black hole case. The key property is that the CMI used is UV finite (cutoff independent), ensuring a robust physical interpretation.

The flux function, constructed this way, decomposes the total entanglement entropy into contributions from every pair of subregions and precisely accounts for both local (intra-boundary) and nonlocal (horizon/spacetime-bridging) entanglement channels.

2. Entanglement Threads versus Bit Threads

Traditional bit thread frameworks (Freedman et al., 2016) rely on maximizing the flux of a divergenceless, norm-bounded vector field through a boundary region to reproduce the Ryu–Takayanagi entropy. This involves density constraints ($|v| \leq 1/4G_N$), and the precise configuration of bit threads is generally non-unique.

Entanglement threads advance this picture by:

• Uniquely fixing thread fluxes through the use of flow equations derived from the entanglement entropies and CMIs of all subregion combinations.
• Tracking how von Neumann entropy is sourced, distinguishing between threads confined to single-side entanglement and those traversing the black hole horizon or wormhole, crossing between asymptotic boundaries.

This refinement provides a detailed map of which elementary pieces share entanglement and how global and local structures collaborate to produce the observed entropy.

3. Tensor Network Interpretation

The entanglement thread construction is closely tied to tensor network concepts. Each entanglement thread is naturally interpreted as a “quantum wire” within a tensor network or circuit representation of the holographic state. In pure AdS/CFT, geodesic trajectories of threads correspond to points in kinematic space.

In the two-sided BTZ black hole, the geometry is mapped to a Poincaré disk where elementary boundary regions of each asymptotic side are mirrored and connected by threads whose fluxes form the network’s “bonds.” This explicit identification provides a microscopic mapping between boundary regions, thread fluxes, and network connections, offering a direct geometric visualization of the underlying entanglement structure and facilitating interpretations in terms of quantum error correction.

4. Perfect-type Entanglement Structure

A notable consequence revealed by this flux-function approach is the emergence of perfect-type entanglement. When the system is divided into regions with both internal (same-side) and wormhole-crossing (opposite-side) entanglement threads, the global state’s entanglement cannot be described as a mere product of decoupled bipartite pairs. Instead, the total state exhibits properties of a perfect tensor (absolutely maximally entangled state): each elementary region is nearly maximally entangled with the rest.

Precisely, when the RT surface transitions (for example, at the Hawking–Page transition), the flux structure dictates that the combination of internal and crossing threads must reassemble into such a maximally entangled, error-correcting code-like state. This observation situates the entanglement thread framework as a microscopic foundation for the emergence of holographic error correction and tensor network dualities, unifying horizon-thread and internal-thread contributions within the perfect tensor paradigm.

5. Relevance to Partial Entanglement Entropy

Partial entanglement entropy (PEE) seeks to attribute a share of the total von Neumann entropy to each elementary region; in the entanglement thread framework, this is implemented by summing the thread fluxes connecting a subregion $A_i$ to the complement: $s_R(A_i) = \sum_{j \notin R} F_{ij} + \rho\,\varpi$ where $R$ is the union of selected elementary regions, $\rho$ is a density factor, and $\varpi$ quantifies the (possibly horizon-crossing) thread contributions in the BTZ context.

This directly realizes the PEE proposal in a thread language and establishes a clear, operational recipe for decomposing entanglement entropy into its microscopic sources. The framework further demonstrates that the PEE is sensitive to both internal structure and wormhole-induced entanglement, and that in the BTZ geometry, horizon-thread contributions can be quantitatively separated.

Summary Table: Key Quantities

Flux Component	Mathematical Expression	Contextual Role
Pairwise Flux ($F_{ij}$)	$½ I(A_i, A_j|\widehat{A})$	Number of threads between subregions
Thread Contribution	$(c/6) \ln(\dots)$	UV-finite, CMI-based flux function calculation
PEE for $A_i$	$\sum_{j \notin R} F_{ij} + \rho\varpi$	PEE as sum over external fluxes + horizon terms

The approach highlights that the flux function of entanglement threads—derived from conditional mutual information—serves as the fundamental object from which von Neumann entropy, PEE, and perfect-type entanglement structures are all composed in a holographic setting. The method’s clarity in distinguishing entanglement sources, its alignment with tensor network models, and its seamless treatment of the BTZ wormhole context mark a significant advance in the microscopic understanding of entanglement structure in quantum gravity (Lin et al., 23 Aug 2025).