  Quantum information spreading and scrambling in many-body systems attract
interests these days. Tripartite mutual information (TMI) based on
operator-based entanglement entropy (EE) is an efficient tool for measuring
them. In this paper, we study random spin chains that exhibit phase transitions
accompanying nontrivial change in topological properties. In their phase
diagrams, there are two types of many-body localized (MBL) states and one
thermalized regime intervening these two MBL states. Quench dynamics of the EE
and TMI display interesting behaviors providing essential perspective
concerning encoding of quantum information. In particular, one of the models is
self-dual, but information spreading measured by the TMI does not respect this
self-duality. We investigate this phenomenon from the viewpoint of spatial
structure of the stabilizers. In general, we find that knowledge of phase
diagram corresponding to qubit system is useful for understanding nature of
quantum information spreading in that system. Connection between the present
work and random circuit of projective measurements and also topological
Majorana quantum memory is remarked.
