An operator product expansion for the mutual information in AdS/CFT

We investigate the behavior of the mutual information IABIAB between two “small” and wide separated spherical regions A and B   in the N=4N=4 SYM gauge theory dual to Type IIB string theory in AdS5×S5AdS5×S5. To this end, the mutual information is recasted in terms of correlators of surface operators W(Σ)W(Σ) defined along a surface Σ   within the boundary gauge theory. This construction relies on the strong analogies between the twist-field operators appearing in the replica trick method used for the computation of the entanglement entropy, and the disorder-like surface operators in gauge theories. In the AdS/CFT correspondence, a surface operator W(Σ)W(Σ) corresponds to having a D3-brane in AdS5×S5AdS5×S5 ending on the boundary along the prescribed surface Σ  . Then, a long distance expansion for IABIAB is provided. The coefficients of the expansion appear as a byproduct of the operator product expansion for the correlators of the operators W(Σ)W(Σ) with the chiral primaries of the theory. We find that, while undergoing a phase transition at a critical distance, the holographic mutual information, instead of strictly vanishing, decays with a power law whose leading contributions of order O(N0)O(N0), originate from the exchange of pairs of the lightest bulk particles between A and B. These particles correspond to operators in the boundary field theory with the smallest scaling dimensions.