  In this paper, we consider the evolution of the thermofield-double state
under the double-traced operator that connects its both sides. We will compute
the entanglement entropy of the resulting state using the replica trick for the
large N field theory. To leading order, it can be computed from the two-point
function of the theory, where, in CFTs, it is fixed by the symmetries. Due to
the exponential decay of the interaction, the entanglement entropy saturates
about the thermal time after the interaction is on. Next, we restrict ourselves
to one dimension and assume that the theory at strong coupling is effectively
described by the Schwarzian action. We then compute the coarse-grained entropy
of the resulting state using the four-point function. The equality of the two
entropies implies that the double-traced operators in our theory act
coherently. In AdS/CFT correspondence where the thermofield-double state
corresponds to a two-sided black hole, the action of a double-traced operator
corresponds to shrinking or expanding the black hole in the bulk.
