We investigate the behaviour of the mutual information I_AB between two "small" and wide separated spherical regions A and B in the N=4 SYM gauge theory dual to Type IIB string theory in AdS_5 × S^5. To this end, the mutual information is recasted in terms of correlators of surface operators 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

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... tion of the entanglement entropy, and the disorder-like surface operators in gauge theories. In the AdS/CFT correspondence, a surface operator W( Σ) corresponds to having a D3-brane in AdS_5 × S^5 ending on the boundary along the prescribed surface Σ. Then, a long distance expansion for I_AB is provided. The coefficients of the expansion appear as a byproduct of the operator product expansion for the correlators of the operators 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(N^0), 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.