Bit threads and holographic entanglement of purification

TL;DR

The paper develops a bit-thread formulation for holographic entanglement of purification (HEoP) using the surface-state correspondence, showing $E_P$ equals the maximum flux through the entanglement wedge neck and connecting this to the entanglement wedge cross section $E_W$ via a generalized MFMC theorem. It also provides a flow-based interpretation of the quantum advantage of dense coding (QAoDC) and proves a monogamy relation with $E_P$ for tripartite states, along with a new lower bound on $S(AB)$ that tightens the Araki–Lieb bound. The framework unifies entanglement measures with thread configurations, enabling direct, operational proofs of EoP properties and new inequalities across $S$, $E_P$, and QAoDC. These results offer a coherent, flow-centric perspective on holographic information processing with potential extensions to multipartite entanglement and horizon contributions in holography.

Abstract

The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a `bit thread' formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for $S(AB)$ in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.

Figures (7)

• Figure 1: Sketch of HEoP. $A, B$ are two non-overlapping regions on the boundary $\partial M$. The entanglement wedge $r_{AB}$ is the region surrounded by $A$, $B$, $A_{opti}'$ and $B_{opti}'$, and $\sigma^{min}_{AB}$ denotes the minimal cross section on $r_{AB}$.
• Figure 2: Sketch of vector field $v_{AB}$ on Riemannian manifold $M$. $A,B$ are the regions on the boundary $\partial M$, and $C=\overline{AB}:=\partial M \backslash (AB)$ is the complement of $AB(\equiv A \cup B)$. The flow $v_{AB}$ is only non-vanishing on $A$ and $B$. Therefore, the flux of any flow from A to B is bounded above by the area of the minimal cross section, the red dashed line as shown in the figure.
• Figure 3: Left: A pure state $|\Psi\rangle_{ABC}$ as the initial purification state of $\rho_{AB}$ that lies on the conformal boundary. Right: The minimal entanglement purification of $\rho_{AB}$, $|\Psi\rangle_{ABC'}$ lying on the boundary of the entanglement wedge $r_{AB}$. To compute $E_{P}(A:B)$, we will restrict the bulk region to the entanglement wedge $r_{AB}$.
• Figure 4: A sketch of a vector field $v_{AB}$ on $r_{AB}$, which is defined as a flow from A to B, whose flux is bounded above by the area of the neck $\sigma^{min}_{AB}$. There is a max flow $\tilde{v}_{AB}$ among all allowed flows $v_{AB}$. Meanwhile, the flux achieves its maximum value $E_{P}(A:B)$.
• Figure 5: Left: The vector field $\tilde{v}_{A(B,C')}$, a flow that simultaneously maximizes the flux $A \rightarrow \bar{A}$ (bounded above by the area of $m_{A}$) and the flux $A\rightarrow B$ (bounded above by the area of $\sigma^{min}_{AB}$). It allows us to compute $S(A)$ and $E_{P}(A:B)$ simultaneously. Right: Similarly, the vector field $\tilde{v}_{B(A,C')}$, a flow that simultaneously maximizes the flux $B \rightarrow \bar{B}$ (bounded above by the area of $m_{B}$) and the flux $B\rightarrow A$ (bounded above by the area of $\sigma^{min}_{AB}$). So we can compute $S(B)$ and $E_{P}(A:B)$ simultaneously.