Disrupting entanglement of black holes

We study entanglement in thermofield double states of strongly coupled conformal field theories by analyzing two-sided Reissner-Nordstrom 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 log E/delta E in units of the temperature. We conjecture that the parametric form of this time scale is universal.