  We examine the dynamics of nearest-neighbor bipartite concurrence and total
correlations in the spin-1/2 $XXZ$ model with random fields. We show, starting
from factorized random initial states, that the concurrence can suffer
entanglement sudden death in the long time limit and therefore may not be a
useful indicator of the properties of the system. In contrast, we show that the
total correlations capture the dynamics more succinctly, and further reveal a
fundamental difference in the dynamics governed by the ergodic versus many-body
localized phases, with the latter exhibiting dynamical oscillations. Finally,
we consider an initial state composed of several singlet pairs and show that by
fixing the correlation properties, while the dynamics do not reveal noticeable
differences between the phases, the long-time values of the correlation
measures appear to indicate the critical region.
