  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.
