Bulk private curves require large conditional mutual information

TL;DR

The paper establishes a link between the existence of private causal curves in AdS bulk and large conditional mutual information in the dual CFT, providing a causal and information-theoretic perspective on entanglement.

Contribution

It proves that private bulk curves imply large conditional mutual information in the boundary theory, connecting causal structures with quantum information measures in AdS/CFT.

Findings

Private curves imply $O(1/G_N)$ conditional mutual information.

Theorem connects bulk causal curves with boundary correlations.

Provides both geometric and information-theoretic proofs.

Abstract

We prove a theorem showing that the existence of "private" curves in the bulk of AdS implies two regions of the dual CFT share strong correlations. A private curve is a causal curve which avoids the entanglement wedge of a specified boundary region $\mathcal{U}$. The implied correlation is measured by the conditional mutual information $I(\mathcal{V}_1:\mathcal{V}_2\,|\,\mathcal{U})$, which is $O(1/G_N)$ when a private causal curve exists. The regions $\mathcal{V}_1$ and $\mathcal{V}_2$ are specified by the endpoints of the causal curve and the placement of the region $\mathcal{U}$. This gives a causal perspective on the conditional mutual information in AdS/CFT, analogous to the causal perspective on the mutual information given by earlier work on the connected wedge theorem. We give an information theoretic argument for our theorem, along with a bulk geometric proof. In the geometric perspective, the theorem follows from the maximin formula and entanglement wedge nesting. In the information theoretic approach, the theorem follows from resource requirements for sending private messages over a public quantum channel.