Seeing the Entanglement Wedge

TL;DR

This work addresses how to reveal the entanglement wedge by semiclassical backreaction, introducing a Connes cocycle flow unitary localized to the causal wedge that can bring the peninsula $\\mathcal{P}[A]$ into causal contact with the boundary for perturbative states with $\mathcal{P}[A] = \mathcal{O}(c G_N)$. The authors demonstrate the mechanism first in Jackiw–Teitelboim gravity with a boundary region comprising disconnected caps, showing that a flow parameter $s$ of order $\log c$ suffices to access Planck-scale proximity to the quantum extremal surface, and then argue for a general perturbative extension, including a boundary modular-flow perspective. They derive the associated stress-energy shocks from the cocycle flow, relate bulk and boundary cocycle implementations, and connect these results to bulk reconstruction and the Gao–Jafferis–Wall paradigm. The findings provide a Lorentzian bulk interpretation of entanglement wedge reconstruction for regions beyond the causal wedge and illuminate how simple bulk operations can, under backreaction, reveal otherwise hidden bulk regions with potential implications for complexity and holographic information recovery.

Abstract

We study the problem of revealing the entanglement wedge using simple operations. We ask what operation a semiclassical observer can do to bring the entanglement wedge into causal contact with the boundary, via backreaction. In a generic perturbative class of states, we propose a unitary operation in the causal wedge whose backreaction brings all of the previously causally inaccessible `peninsula' into causal contact with the boundary. This class of cases includes entanglement wedges associated to boundary sub-regions that are unions of disjoint spherical caps, and the protocol works to first order in the size of the peninsula. The unitary is closely related to the so-called Connes Cocycle flow, which is a unitary that is both well-defined in QFT and localised to a sub-region. Our construction requires a generalization of the work by Ceyhan & Faulkner to regions which are unions of disconnected spherical caps. We discuss this generalization in the Appendix. We argue that this cocycle should be thought of as naturally generalizing the non-local coupling introduced in the work of Gao, Jafferis & Wall.

Figures (7)

• Figure 1: Left: given a boundary region $A$, the RT surface $\mathcal{R}_{A}$ and the corresponding entanglement wedge $\mathcal{W}_{E}[A]$(with black boundary), the outermost extremal surface $\mathcal{R}'_{A}$ and the corresponding outermost wedge $\mathcal{W}_{O}[A]$ (with orange boundary) is shown. The green dotted lines mark the causal horizons and therefore the boundaries of $\mathcal{W}_{C}[A]$. The intersection of the horizons defines the causal surface $\mathcal{C}$. Right: We consider bulk perturbations that don't change the state in $\mathcal{W}_{\overline{C}}[A]$, but change the geometry so as to bring parts of $\mathcal{W}_{E}[A] \cap \mathcal{W}_{\overline{C}}[A]$ into the causal past of $D(A)$ as shown. A signal (marked by the black straight arrow) can then reach $D(A)$. The causal past of $D(A)$ cannot be extended beyond $\mathcal{R}'_{A}$ under this class of operations. The peninsula $\mathcal{P}[A]$, marked in orange, is therefore the largest region that can be placed in the causal past (or future) of $D(A)$ using such operations, while the red region can never be accessed this way.
• Figure 2: Region $a=a_{1}\cup a_{2}$ is shown in orange. Each $D(a_{r})$ is an AdS-Rindler wedge whose future and past boundary lies on a Killing horizon. For $D(a_{2})$, the horizons are marked by $x^-_2=0$ and $x^+_2=0$ respectively.
• Figure 3: The green region marks $a(X^{+}_2)$ which is a deformation of $a$ by moving $\partial a_2$ in the null direction along $x^-_2=0$. The $\psi_s$ states have stress tensor shocks at $\partial a$ proportional to the derivative of the von Neumann entropy under such shape deformations.
• Figure 4: We consider the region in the dual theory $A = A_1 \cup A_2$. In this figure, $A_1$ and $A_2$ are overlaid on the dual picture. The relevant regions in the gravity theory $a_1$ and $a_2$ are also shown.
• Figure 5: We act with the two-sided unitary $u_s(\Omega, a_1 \cup a_2)$ on the vacuum as in \ref{['eqn:psis']}. This has the effect of moving the right boundary particle's trajectory to the dashed green line so that the whole peninsula $\mathcal{P}$ is in the past of the right boundary particle's worldline, to leading order in large $c$. The shocks are labeled by the green arrows. For the purposes of seeing the peninsula, the left-most shocks are irrelevant, although they will have an effect on the left boundary particle's trajectory, which is not shown. Also note that the regions denoted $a_1, a_2$ are the $s=0$ positions of the regions. Note that even though $A_1$, $A_2$ are defined as fixed regions in the baths, their coordinate positions change in the picture after we act with $u_s$. We caution that the main text only works to leading order perturbatively and so we have exaggerated these effects in this figure.