Disrupting entanglement of black holes

We study entanglement in thermofield double states of strongly coupled conformal field theories by analyzing two-sided Reissner-Nordstr\"om solutions in anti--de Sitter space. The central object of study is the mutual information between a pair of regions, one on each asymptotic boundary of the black hole. For large regions the mutual information is positive and for small ones it vanishes; we compute the critical length scale, which goes to infinity for extremal black holes, of the transition. We also generalize the butterfly effect of Shenker and Stanford J. High Energy Phys. 03 (2014) 067 to a wide class of charged black holes, showing that mutual information is disrupted upon perturbing the system and waiting for a time of order logE/ in units of the temperature. We conjecture that the parametric form of this time scale is universal.