Entanglement Structure of Nonlocal Field Theories

TL;DR

The paper addresses how strong nonlocality in a bosonic field theory alters entanglement beyond entropy, focusing on mutual information and tripartite information. It combines a lattice, Gaussian-state analysis with nonlocal kernels and a holographic RT-based dual to compare correlation structures across viewpoints. Key findings show that the nonlocality scale $A$ induces volume-law entanglement for small subsystems and produces long-range, monogamous multipartite entanglement in the field theory, yet the holographic dual predicts a suppression of both mutual and tripartite information in the volume-law phase. This tension suggests that nonlocal quantum states require holographic descriptions beyond classical geometry to fully capture their intricate entanglement networks and calls for non-geometric extensions of holography.

Abstract

Nonlocal interactions are known to generate volume-law entanglement entropy. However, their deeper impact on the fine structure of quantum correlations remains a key open question. In this work, we explore a bosonic nonlocal field theory, examining correlation measures beyond entanglement entropy, namely, mutual information and tripartite information. Using numerical lattice simulations, we show that the nonlocality scale, \(A\), not only determines the onset of volume-law behavior but also leads to striking features: notably, extremely long-range mutual information and an unusual monogamy structure. In this regime, increasing the separation between large regions can paradoxically enhance their multipartite entanglement. Through holographic duality, we verify that the Ryu-Takayanagi formula correctly captures the volume-law scaling of entropy. Yet, a significant tension emerges: while the field theory reveals rich spatial correlations, the holographic model predicts a complete suppression of both mutual and tripartite information in the volume-law phase. This non-monogamous behavior in the holographic description stands in sharp contrast to the monogamous and highly structured entanglement observed in the field theory. Our results demonstrate that nonlocality gives rise to quantum states of such complexity that conventional geometric models of spacetime fall short. This points to the need for a new framework that goes beyond geometry to fully capture the nature of these correlations.

Figures (5)

• Figure 1: Entanglement entropy $S_\Omega$ as a function of subsystem length $l$ for varying nonlocality parameters $A$. Left: For moderate nonlocality ($A = 40, 60, 80$), the entropy exhibits a volume-law scaling regime for small $l$, with the extent of this regime expanding as $A$ increases. Right: For large nonlocality ($A = 400, 600, 800, 1000$), the volume-law scaling is significantly enhanced and persists over a much wider range of $l$, demonstrating the profound impact of strong nonlocality on the entanglement structure.
• Figure 2: Mutual information between two disjoint intervals of equal length ($l_1 = l_2 = 10$) as a function of the normalized separation $x/L$ where $L=l_{1}+ l_{2}$ is the total length of subsystems. Left: For moderate nonlocality parameters ($A = 40, 60, 80$), the mutual information exhibits a gradual decay with separation, highlighting the persistence of correlations at large distances. Right: For strong nonlocality ($A = 400, 600, 800$), the mutual information remains substantial across a wide range of separations, confirming that strong nonlocality drastically slows the decay of quantum correlations.
• Figure 3: Tripartite information $I_3$ for three disjoint regions as a function of their separation. Left: For small, equal-length regions, the tripartite information is consistently negative, confirming the monogamous nature of the entanglement. Right: For larger, equal-length regions, the negative values become significantly more pronounced with separation, especially for higher $A$.
• Figure 4: Holographically computed minimal surface area (proportional to entanglement entropy) for $d=2$, $\omega=1$ as a function of subsystem size $l$. Left: For small nonlocality parameters ($A = 25, 30, 35$), the area exhibits a clear linear scaling with $l$ in the regime $l \ll A$, confirming a volume law. Right: For a broader range of parameters from moderate to large ($A = 40, 60, 400$), the volume-law scaling persists, and the slope of the linear growth is significantly enhanced with increasing $A$, demonstrating its role in amplifying entanglement.
• Figure 5: Holographic mutual information (left) and tripartite information (right) for $d=2$, $\omega=1$, with equal subsystem lengths $l=50$, plotted against normalized separation $x/L$ for $A=40,60,400$. As $A$ increases, both mutual information and tripartite information undergo dramatic suppression. For strong nonlocality ($A=400$), mutual information and tripartite information almost vanishes for all separations.