Causality & holographic entanglement entropy

TL;DR

This work establishes that the covariant holographic entanglement entropy prescription (HRT) is consistent with relativistic causality provided the bulk obeys the null energy condition. It introduces and formalizes the entanglement wedge ${\cal W}_{E}[{\cal A}]$ as the natural bulk region associated with a boundary region ${\cal A}$ and shows that the extremal surface ${\cal E}_{\cal A}$ computing ${S_{\cal A}}$ must reside in the causal shadow ${\cal Q}_{\partial {\cal A}}$, thereby preserving boundary causality. The authors develop a bulk proof using null geodesic congruences in AdS$_{3}$ and a general theorem that relates bulk causal structure to boundary decomposition, strengthening evidence for the HRT proposal and clarifying the role of a spacelike homology condition. They discuss implications for the dual of the reduced density matrix ${\rho_{\cal A}}$, potential extensions to higher-derivative and quantum corrections, and the broader impact on holographic causality and bulk reconstruction. Overall, the paper provides a rigorous causality check for holographic entanglement entropy, introduces the entanglement wedge as a natural dual object, and outlines the conditions under which HRT remains valid and meaningful in dynamical spacetimes.

Abstract

We identify conditions for the entanglement entropy as a function of spatial region to be compatible with causality in an arbitrary relativistic quantum field theory. We then prove that the covariant holographic entanglement entropy prescription (which relates entanglement entropy of a given spatial region on the boundary to the area of a certain extremal surface in the bulk) obeys these conditions, as long as the bulk obeys the null energy condition. While necessary for the validity of the prescription, this consistency requirement is quite nontrivial from the bulk standpoint, and therefore provides important additional evidence for the prescription. In the process, we introduce a codimension-zero bulk region, named the entanglement wedge, naturally associated with the given boundary spatial region. We propose that the entanglement wedge is the most natural bulk region corresponding to the boundary reduced density matrix.

Figures (11)

• Figure 1: For AdS$_{3}$, the RT formula satisfies field-theory causality marginally. The plane generated by null geodesics (color-coded by angular momentum) from a given boundary point (blue) is also ruled by spacelike geodesics at constant time (color-coded by time).
• Figure 2: An illustration of the causal domains associated with a region ${\cal A}$, making manifest the decomposition of the spacetime into the four distinct domains indicated in \ref{['bdy4d']}. Two deformations ${\cal A}'$ are also included for illustration in the right panel.
• Figure 3: Example of a causally trivial spacetime and a boundary region ${\cal A}$ whose causal shadow is a finite spacetime region. We have engineered an asymptotically AdS$_{3}$ geometry sourced by matter satisfying the null energy condition (see footnote \ref{['fn:metric']}) and taken ${\cal A}$ to nearly half the boundary, $\varphi_{\cal A} = 1.503$, at $t=0$ (thick red curve). The shaded regions on the boundary cylinder are $D[{\cal A}]$ and $D[{\cal A}^c]$ respectively. The extremal surface is the thick blue curve, while the purple curves are the rims of the causal wedge (causal information surfaces) for ${\cal A}$ and ${\cal A}^c$ respectively. A few representative generators are provided for orientation: the blue null geodesics generate the boundary of the causal wedge for ${\cal A}$ while the green ones do likewise for ${\cal A}^c$. The orange generators in the middle of the spacetime generate the boundary of the causal shadow region ${\cal Q}_{ \partial {\cal A}}$.
• Figure 4: Sketch of Penrose diagram for (a) static eternal Schwarzschild-AdS$_{}$ and (b) 'thin shell' Vaidya-Schwarzschild-AdS$_{}$, with the various regions labeled. The AdS boundaries are represented by vertical black lines, the singularities by purple curves, the horizons by diagonal blue lines, and the 'shell' in the Vaidya case by diagonal brown line.
• Figure 5: Sketch of Penrose diagram for a symmetric Vaidya-Schwarzschild-AdS$_{}$ geometry obtained by imploding null shells to the past and future from both boundaries. The crucial new feature of note is the presence a causal shadow region that is spacelike separated from both boundaries. We have also indicated the extremal surface ${\cal E}_{\cal A}$ for the region ${\cal A} = \Sigma_R$ in red at the center of the figure and ${\cal F}_{\cal A}$ is a ${\bf S}^{d-1}$ of finite area in the causal future of the left boundary. The lightly shaded regions are the causal wedges associated with ${\cal A}$ and ${\cal A}^c$ respectively.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Corollary 3
• Corollary 4
• Lemma 5
• Lemma 6
• Theorem 7
• Corollary 8
• Lemma 9
• Lemma 10