Entanglement of Purification and Multiboundary Wormhole Geometries

TL;DR

The paper develops a geometric framework for entanglement of purification (EP) in holographic CFTs, extending to multipartite cases and linking EP to the entanglement wedge cross section E_W in AdS3. By exploiting multiboundary wormhole constructions via AdS3 identifications, it derives new holographic inequalities for E_W and translates them into EP inequalities, while also showing how EP calculations can be performed via wormhole entropies in vacuum AdS3. A key proposal is that, under the E_P=E_W conjecture, the wormhole geometry provides an optimal purification with minimal Hilbert space, with purifying degrees of freedom localized on RT-surface segments and boundaries closely corresponding to purified subsystems. The work establishes a calculational bridge between EP and RT entropies, reveals a rich set of constraints on holographic states, and points to future extensions in higher dimensions and potential field-theoretic proofs of the EP–EW correspondence.

Abstract

We posit a geometrical description of the entanglement of purification for subregions in a holographic CFT. The bulk description naturally generalizes the two-party case and leads to interesting inequalities among multi-party entanglements of purification that can be geometrically proven from the conjecture. Further, we study the relationship between holographic entanglements of purification in locally-AdS3 spacetimes and entanglement entropies in multi-throated wormhole geometries constructed via quotienting by isometries. In particular, we derive new holographic inequalities for geometries that are locally AdS3 relating entanglements of purification for subregions and entanglement entropies in the wormhole geometries.

Figures (12)

• Figure 1: The entanglement wedge cross section of the $AB$ boundary subsystem, with size $E_W$. This object, depicted as a red line, is the minimal surface that totally partitions the entanglement wedge into a region adjacent to $A$ and one adjacent to $B$. It can be extended to a boundary-anchored geodesic (red dashed line), which further partitions the complement of $A \cup B$ into $P_A$, $P_B$, $Q_A$, and $Q_B$.
• Figure 2: The tripartite entanglement wedge cross section (red line) of the $ABC$ boundary subsystem, with area $E_W$. Note that the minimization here is a constrained minimization such that the red line must separate $A$, $B$, and $C$ from their respective complements in the entanglement wedge and be connected.
• Figure 3: A partition of $n = 7$ boundary subregions into $n_1 = 3$ sets of subregions and $n_2 = 4$ sets of subregions. The surface $P_1$, which computes the tripartite entanglement of purification for the set $C_1 = \{A_1 A_2 A_3 A_4, A_5, A_6 A_7\}$, is shown in red, and the surface $P_2$, which computes the quadripartite entanglement of purification for the set $C_2 = \{A_2, A_3, A_4 A_5 A_6, A_7 A_1\}$, is shown in blue. Segments of these two surfaces form the polygon $\bar{P}$, shown with the dashed purple line, which can be deformed to the 7-partite entanglement wedge cross section (shown in solid purple).
• Figure 4: A partition of $n = 7$ boundary subregions into $n_1 = 2$ sets of subregions and $n_2 = 5$ sets of subregions. $P_1$ (now a minimal 2-gon) is shown in red and $P_2$ is shown in blue. As before, $\bar{P}$ is shown in dashed purple, and the surface that computes $E_W(A_1:\ldots:A_7)$ is shown in solid purple.
• Figure 5: A partition of $n = 7$ boundary subregions into $n_1 = 2$ sets of subregions, $n_2 = 2$ sets of subregions, and $n_3 = 3$ sets of subregions.