  Isolated quantum systems at strong disorder can display many-body
localization (MBL), a remarkable phenomena characterized by an absence of
conduction even at finite temperatures. As the ratio of interactions to
disorder is increased, one expects that an MBL phase will eventually undergo a
dynamical phase transition to a delocalized phase. Here we constrain the nature
of such a transition by exploiting the strong subadditivity of entanglement
entropy, as applied to the many-body eigenstates close to the transition in
general dimensions. In particular, we show that at a putative continuous
transition between an MBL and an ergodic delocalized phase, the critical
eigenstates are necessarily thermal, and therefore, the critical entanglement
entropy equals the thermal entropy. We also explore a qualitatively different
continuous localization-delocalization transition, where the delocalized phase
is non-ergodic whose volume law entanglement entropy tends to zero as the
transition is approached.
