  We probe the multipartite entanglement structure of the vacuum state of a CFT
in 1+1 dimensions, using recovery operations that attempt to reconstruct the
density matrix in some region from its reduced density matrices on smaller
subregions. We use an explicit recovery channel known as the twirled Petz map,
and study distance measures such as the fidelity, relative entropy, and trace
distance between the original state and the recovered state. One setup we study
in detail involves three contiguous intervals $A$, $B$ and $C$ on a spatial
slice, where we can view these quantities as measuring correlations between $A$
and $C$ that are not mediated by the region $B$ that lies between them. We show
that each of the distance measures is both UV finite and independent of the
operator content of the CFT, and hence depends only on the central charge and
the cross-ratio of the intervals. We evaluate these universal quantities
numerically using lattice simulations in critical spin chain models, and derive
their analytic forms in the limit where $A$ and $C$ are close using the OPE
expansion. In the case where $A$ and $C$ are far apart, we find a surprising
non-commutativity of the replica trick with the OPE limit. For all values of
the cross-ratio, the fidelity is strictly better than a general
information-theoretic lower bound in terms of the conditional mutual
information. We also compare the mutual information between various subsystems
in the original and recovered states, which leads to a more qualitative
understanding of the differences between them. Further, we introduce
generalizations of the recovery operation to more than three adjacent
intervals, for which the fidelity is again universal with respect to the
operator content.
