  The mutual information of disconnected regions in large $N$ gauge theories
with holographic gravity duals can undergo phase transitions. These occur when
connected and disconnected bulk Ryu-Takayanagi surfaces exchange dominance.
That is, the bulk `soap bubble' snaps as the boundary regions are drawn apart.
We give a gauge-theoretic characterization of this transition: States with and
without a certain defect operator insertion -- the defect separates the
entangled spatial regions -- are shown to be perfectly distinguishable if and
only if the Ryu-Takayanagi surface is connected. Meanwhile, states with and
without a certain Wilson loop insertion -- the Wilson loop nontrivially threads
the spatial regions -- are perfectly distinguishable if and only if the
Ryu-Takayanagi surface is disconnected. The quantum relative entropy of two
perfectly distinguishable states is infinite. The results are obtained by
relating the soap bubble transition to Hawking-Page (deconfinement) transitions
in the Renyi entropies, where defect operators and Wilson loops are known to
act as order parameters.
