  Chaotic transverse-field Ising model with measurements exhibits interesting
purification dynamics. Ensemble of non-unitary dynamics of a chaotic many-body
system with measurements exhibits a purification phase transition. We
numerically find that the law of the increase dynamics of the purity changes by
projective measurements in the model. In order to study this behavior in
detail, we construct the formalism of the tripartite mutual information (TMI)
for non-unitary time evolution operator by using the state-channel map. The
numerical result of the saturation value of the TMI indicates the existence of
a measurement-induced phase transition. This implies the existence of two
distinct phases, mixed phase and purified phase. Furthermore, the real-space
spread of the TMI is investigated to explore spatial patterns of information
spreading. Even in the purified phase, the spatial pattern of the light cone
spread of quantum information is not deformed, but its density of information
propagation is reduced on average by the projective measurements. We also find
that this spatial pattern of the TMI distinguishes the chaotic and integrable
regimes of the system.
