Entanglement and correlations between local observables in de Sitter spacetime

TL;DR

This work develops a local Gaussian-information approach to entanglement in the cosmological patch of de Sitter space, using smeared, compactly supported local modes and a phase-space geometry encoded by a covariance metric σ and a complex structure J. The Bunch-Davies vacuum induces curvature-dependent, nearly scale-invariant correlations that grow with the Hubble rate, but paradoxically reduce the entanglement between two local degrees of freedom, as quantified by logarithmic negativity. The authors introduce partner modes to describe how entanglement is distributed spatially, showing that curvature pushes entanglement into nonlocal, non-accessible degrees of freedom, while local mode entropy increases due to strong coupling to their partners. These results reconcile entropy-based and entanglement-harvesting perspectives, emphasize the distinction between correlation and entanglement in curved spacetime, and have implications for the interpretation of inflationary perturbations and their observable signatures.

Abstract

Studies of quantum field entanglement in de Sitter space based on the von Neumann entropy of local patches have concluded that curvature enhances entanglement between regions and their complements. Similar conclusions about entanglement enhancement have been reached in analyses of Fourier modes in the cosmological patch of de Sitter space. We challenge this interpretation by adopting a fully local approach: examining entanglement between pairs of field modes compactly supported within de Sitter's cosmological patch. Our approach is formulated in terms of the properties of a metric tensor and an associated complex structure induced by the Bunch-Davies vacuum on the classical phase space. We find that increasing curvature increases correlations between local modes but, somewhat counterintuitively, decreases their entanglement. Our methods allow us to characterize how entanglement is spatially distributed, revealing that a cosmological constant, even if tiny, qualitatively alters the vacuum's entanglement structure. We show that our results are compatible with previous entropy-based studies when properly interpreted. Our findings have implications for entanglement between observables generated during cosmic inflation.

Figures (11)

• Figure 1: Shape of the smearing functions $f^{(\delta)}(\vec{x})$ for a few values of $\delta$ (with $R=1$). $r$ represents the radial coordinate.
• Figure 2: (a) Field–field, (b) momentum–momentum, and (c) field–momentum smeared self-correlations in the Bunch-Davies vacuum for the single mode defined in Eq. \ref{['eq:ex1vNS']}. Note the growth of the field–field correlations as $\mu$ decreases—this growth signals an infrared divergence in the limit $\mu \to 0$. The momentum-momentum and field-momentum correlations are only mildly dependent on $\mu$ so that these lines appear on top of each other. The dashed lines represent the terms $\mathcal{N}_{\Phi\Phi}$, $\mathcal{N}_{\Pi\Pi}$, and $\mathcal{N}_{\Phi\Pi}$ in Eqs. \ref{['phiphi']}--\ref{['phipi']}. These figures confirm that these terms are subleading in the regime of interest ($RH \gg 1$ and $\mu \ll 1$), and they will therefore be neglected from now on.
• Figure 3: (a) Geometric representation of the support of the mode in Eq. \ref{['eq:ex2vNS']}. (b) The smearing function $f_S(r)$ as a function of the radial distance $r=|\vec{x}|$. The support of the shell is shaded in orange.
• Figure 4: (a) Field–field, (b) momentum–momentum, and (c) field–momentum correlations between two identical modes supported in non-overlapping balls, shown as functions of their separation $|\Delta \vec{x}|$, for $\mu^2 = 10^{-4}$ and various values of $RH$. The modes are defined as in Example \ref{['ex:singlemodeball']} with $\delta = 2$. The non-overlapping condition $|\Delta \vec{x}| > 2R$ explains why the lines corresponding to different values of $R$ are plotted over different ranges of $|\Delta \vec{x}|$. Panel (a) illustrates the almost scale-invariant behavior of the field–field correlations when $|\Delta \vec{x}| H \gtrsim 1$. In contrast, the momentum-momentum and field-momentum correlations are not almost scale-invariant
• Figure 5: Von Neumann entropy of a single-mode subsystem supported in a ball of radius $R$, with the mode defined in Eq. \ref{['eq:ex1vNS']}. The entropy is shown for different values of the mass parameter $\mu$. This figure shows that $S(\nu_A)$ grows with $H$, except when the region supporting the mode is small compared to the Hubble radius ($RH \ll 1$). In this regime, the entropy becomes independent of $H$ and converges to its Minkowski value. Additionally, the value of $S(\nu_A)$ increases with decreasing mass, as expected due to the infra-red divergence in the massless limit.

Theorems & Definitions (33)

• Proposition 1
• Example 1
• Example 2
• Example 3
• Proposition 2
• proof
• Example 4
• Example 5
• Example 6
• Proposition 3