  We examine the interplay of interaction and disorder for a Heisenberg spin
ladder system with random fields. We identify many-body localized states based
on the entanglement entropy scaling, where delocalized and localized states
have volume and area laws, respectively. We first establish the quantum phase
transition at a critical random field strength $h_c \sim 8.5\pm 0.5$, where all
energy eigenstates are localized beyond that value. Interestingly, the
entanglement entropy and fluctuation of the bipartite magnetization show
distinct probability distributions which characterize different quantum phases.
Furthermore, we show that for weaker $h$, energy eigenstates with higher energy
density are delocalized while states at lower energy density are localized.
This defines a mobility edge and a mobility gap separating these two phases. By
following the evolution of low energy eigenstates, we observe that the mobility
gap grows with increasing the random field strength, which drives the system to
the phase of the full many-body localization with increasing disorder strength.
