Holographic mutual information and distinguishability of Wilson loop and defect operators

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.