Holographic multipartite entanglement structures in IR modified geometries

TL;DR

This paper develops two IR-modified holographic geometries (spherical and hyperbolic) to explore how long-range multipartite entanglement on the boundary reorganizes across scales. By leveraging entanglement wedge cross sections (EWCS), multi-EWCS, and multi-entropy, it reveals that spherical deformations push multipartite entanglement toward longer scales, saturating upper information-theoretic bounds, while hyperbolic deformations suppress long-range multipartite correlations and can realize triangle/polygon-state structures with minimal EWCS. The work introduces new multipartite signals and analyzes extremal values under quantum-marginal constraints, offering concrete interpretations for measures like the Markov gap, L-entropy, and a novel Upsilon quantity. Overall, IR geometry acts as a practical tool to realize extremal entanglement regimes and to study quantum marginal problems within holography, with potential implications for entanglement classification and quantum information constraints in holographic theories.

Abstract

We investigate how IR modifications of the bulk geometry reshape long-range multipartite entanglement on the boundary in holography. We modify the IR geometries in two opposite directions: spherical modifications that enhance long-range entanglement and hyperbolic modifications that suppress them. We utilize various multipartite entanglement measures/signals to analyze the multipartite entanglement structures. These measures/signals are combinations of entanglement entropy, multi-entropy, entanglement wedge cross sections (EWCS) and multi-EWCS. Our results reveal that in the extremal limits of these two geometric modifications, the multipartite entanglement structures exhibit starkly contrasting behaviors: various measures saturate either their theoretical upper or lower bounds in the respective geometries. This demonstrates that IR deformations provide a practical holographic framework for realizing extremal entanglement regimes. Moreover, it serves as an effective tool for studying quantum marginal problems in holography. Finally, by observing how different measures respond to these engineered geometries, we gain clarifying insights into the specific types of multipartite entanglement that each measure/signal is particularly sensitive to.

Figures (28)

• Figure 1: A graphical summary of the geometries in Ju_2024 with modified IR regions. The left figure depicts the general case where the geometry of the IR region (shown in yellow) is modified in a global AdS$_3$ spacetime. Displayed in purple is the edge of this IR region, where matter fields reside and spatial connection conditions are imposed. The middle and the right figures depict the diametrically opposite toy-model geometries obtained through such modification. The middle figure shows the spherical extremal case, where the IR region with large positive curvature (shown in red) can be viewed as a hemisphere embedded in an imaginary Euclidean space. The right figure, on the other hand, shows the hyperbolic extremal case, where the IR region with extremely negative curvature (shown in blue) infinitely approaches a light cone embedded in an imaginary Minkowski spacetime.
• Figure 2: The RT surfaces in the spherical (left) and hyperbolic (right) extremal cases. As in figure \ref{['modIR']}, the Cauchy slices of the spherical and hyperbolic extremal geometries are embedded respectively in higher-dimensional imaginary Euclidean and Minkowski backgrounds. The left and right figure depict respectively the four RT surface phases in the spherical extremal case and the three phases in the hyperbolic extremal case, along with their corresponding boundary regions.
• Figure 3: The vanishing-CMI configurations for the spherical (left) and hyperbolic (right) extremal cases. Specifically, the figures plot the Cauchy slices of the spherical and hyperbolic extremal geometries in figure \ref{['CMI']} in stereographic projection. The left figure represents the vanishing CMI for the spherical extremal case: the RT surfaces for $AE$ (purple), $BE$ (red), $E$ (blue), and $ABE$ (green) are shown, and their contributions cancel exactly in the CMI combination. An analogous cancellation occurs in the hyperbolic extremal case, as illustrated in the right figure.
• Figure 4: Entanglement structures are depicted using threads representing the entanglement (CMI between two infinitesimal subregions) between the two points they connect. The middle figure displays the entanglement in vacuum AdS, with entanglement at all length scales. On the left the spherical extremal case is shown with all $L > L_c$ long-scale entanglement eliminated and transferred to the longest scale $L = \pi l_{S^1}$. On the right figure, in the hyperbolic extremal case, all $L > L_c$ long-scale entanglement is eliminated and transferred to the critical length $L = L_c$.
• Figure 5: The calculation of EWCS in the pure AdS and the modified geometries. The entire conformal boundary is partitioned into three equal subregions $A$, $B$, and $C$. In the left, middle, and right figures, the purple curves represent $\gamma'_{A,C}$ in the pure AdS, the modified spherical geometry, and the modified hyperbolic geometry, respectively. The three blue curves denote the minimal surfaces homologous to $A$, $B$, and $C$. The modified IR region is sufficiently small so that the entanglement wedges of $A$, $B$, and $C$ remain unchanged after the geometric modification.

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• Lemma 1
• Theorem 3
• Lemma 2
• Lemma 3
• Theorem 4