  We analyze the saturation value of the bipartite entanglement and number
entropy starting from a random product state deep in the MBL phase. By studying
the probability distributions of these entropies we find that the growth of the
saturation value of the entanglement entropy stems from a significant
reshuffling of the weight in the probability distributions from the bulk to the
exponential tails. In contrast, the probability distributions of the saturation
value of the number entropy are converged with system size, and exhibit a sharp
cut-off for values of the number entropy which correspond to one particle
fluctuating across the boundary between the two halves of the system. Our
results therefore rule out slow particle transport deep in the MBL phase and
confirm that the slow entanglement entropy production stems uniquely from
configurational entanglement.
