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Bit threads and holographic entanglement of purification

Jonathan Harper, Matthew Headrick

TL;DR

This work extends the bit-thread formalism to holographic entanglement of purification by formulating dual flow programs via convex duality for both bipartite and multipartite cases. It proves key inequalities by treating E_w as a constrained max-flow problem, showing how restricting to the homology region or adding boundary constraints limits flux and yields bounds that align with entropic quantities like $S$, $I$, and $E_p$. In the multipartite setting, the authors introduce a coupled flow structure with a boundary function $oldsymbol{α}$ on a shared surface $oldsymbol{O}$, decomposing the total flux into bipartite contributions and a genuinely multipartite residue, captured by the $oldsymbol{α}$-flux. The results suggest a decomposition $E_p(oldsymbol{A}) = ext{bipartite sum} + M(oldsymbol{A})$, provide a framework for covariant and distillation-related extensions, and outline avenues toward a rigorous proof that holographic $E_p$ equals the bit-thread dual $E_w$ under suitable conditions.

Abstract

Generalizing the bit thread formalism, we use convex duality to derive dual flow programs to the bipartite and multipartite holographic entanglement of purification proposals and then prove several inequalities using these constructions. In the multipartite case we find the flows exhibit novel behavior which allows for a constrained flux on the boundary of the homology region. We show this flux can be made distinct from bipartite terms and reflects the truly multipartite portion of the holographic entanglement of purification.

Bit threads and holographic entanglement of purification

TL;DR

This work extends the bit-thread formalism to holographic entanglement of purification by formulating dual flow programs via convex duality for both bipartite and multipartite cases. It proves key inequalities by treating E_w as a constrained max-flow problem, showing how restricting to the homology region or adding boundary constraints limits flux and yields bounds that align with entropic quantities like , , and . In the multipartite setting, the authors introduce a coupled flow structure with a boundary function on a shared surface , decomposing the total flux into bipartite contributions and a genuinely multipartite residue, captured by the -flux. The results suggest a decomposition , provide a framework for covariant and distillation-related extensions, and outline avenues toward a rigorous proof that holographic equals the bit-thread dual under suitable conditions.

Abstract

Generalizing the bit thread formalism, we use convex duality to derive dual flow programs to the bipartite and multipartite holographic entanglement of purification proposals and then prove several inequalities using these constructions. In the multipartite case we find the flows exhibit novel behavior which allows for a constrained flux on the boundary of the homology region. We show this flux can be made distinct from bipartite terms and reflects the truly multipartite portion of the holographic entanglement of purification.

Paper Structure

This paper contains 44 sections, 3 theorems, 102 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Bipartite entanglement of purification
  3. Flows and relative homology
  4. Flow formulation
  5. Relaxation of the homology region
  6. Adding constraints
  7. Conditional EOP
  8. Bounds
  9. $\frac{1}{2}I(A:B) \leq E_{w}(A:B) \leq \min[S(A),S(B)]$
  10. Upper bound
  11. Lower bound
  12. Saturation
  13. $E_{w}(A:BC) \geq E_{w}(A:B)$
  14. $E_{w}(A\tilde{A}:B\tilde{B}) \geq E_{w}(A:B) + E_{w}(\tilde{A}:\tilde{B})$
  15. Saturation
  16. ...and 29 more sections

Key Result

Theorem 1

Let $P$ be a convex maximization program: with solution $y^{*} \in \mathcal{F}$ where $\mathcal{F}$ is the set of feasible points. Let $\tilde{P}$ be a second convex maximization program obtained by imposing an additional set of constraints $g_{m}(y) \leq0 \ \forall m \ and \ l_{n}(y) =0 \ \forall n$ on $P$: and similarly define $\tilde{y}^{*} \in \tilde{\mathcal{F}}$. Then $f_{0}(y^{*}) \geq f_

Figures (26)

  • Figure 1: Illustration of the homology-region cross section $E_w(A:B)$, conjectured to be dual to the entanglement of purification \ref{['EOPdef']}. The $AB$ homology region $r(AB)$ is shown in blue. $E_w(A:B)$ is defined as the area of the minimal surface $b_p(A:B)$ in $r(AB)$ homologous to $A$ relative to $m(AB)$.
  • Figure 2: The holographic multipartite EOP proposal of Umemoto2018, illustrated for the case of three regions. The joint RT surface $\mathcal{O}\coloneqq m(\mathcal{A})$ is shown in red. This is partitioned into regions $A_i'$, and the total area of the corresponding minimal surfaces $m(A_iA_i')$ is minimized over partitions. The minimal surfaces $\Sigma(A_{i})$ are shown as colored dashed lines; their union is $\Sigma$. Also shown as a dotted line is the minimal surface $b_p(A_1)$ which computes the bipartite EOP $E_w(A_1)$.
  • Figure 3: Shrinking $M$ to $M'$.
  • Figure 4: A maximal flow of \ref{['floweop']} whose flux calculates $E_{w}(A:B)$.
  • Figure 5: Left: The flow $x^{\mu}$ acts to change the geometry $v^{\mu}$ can probe. Right: On the reduced manifold the maximum flux of $v^{\mu}$ gives the holographic EOP.
  • ...and 21 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Definition 1: Multiflow
  • Definition 2: Subflow
  • Theorem 2: Existence of a maximal multiflow
  • Theorem 3: Monogamy of mutual information (MMI)
  • proof