Information transfer between two black holes in JT gravity

We study the model of two 2d AdS black holes connected to each other via a reservoir, represented by a flat strip. The quantum degrees of freedom of the reservoir are described by a 2d CFT which we assume to be holographic. Using the quantum extremal surface prescription, we compute the mutual information between the two black holes and the tripartite information of the black holes with a region in the reservoir. We show that there is a bound on the reservoir size, below which the black holes can share the mutual information. At the transition point the entanglement wedge of the two black holes jumps, and we show that there is a region inside of each black hole spacetime which can only be reconstructed by the holographic duals of both black holes. The computation of the tripartite information allows to extend this reconstruction to arbitrary reservoir size, at the cost of adding degrees of freedom from the reservoir to the reconstructing operator algebra.