The combined effect of unitary quantum dynamics and quantum measurement backaction leads to the emergence of unique phenomena, like the Quantum Zeno effect. Long limited to the case of single particles and few-body systems, the study of measurement-induced dynamics has recently come under much scrutiny for quantum-measured many-body systems, leading to the discovery of measurement-induced phase transitions (MiPTs). As a newly discovered out-of-equilibrium phase transition, it has drawn broad cross-disciplinary works, ranging from condensed matter physics and statistical mechanics to quantum information, quantum computation, and error correction, with several studies characterising its features in different models and scenarios. In this thesis, we address a general yet subtle feature of MiPTs: how does partial information, an incomplete set of measurement outcomes, affect the behaviours of these phase transitions? We address various facets of incomplete observer's information. We first consider the case of imperfect detection via a model of inefficient measurement, in which part of the information is lost, resulting in a density-matrix description of the system's state of knowledge. Inefficiency introduces different phase transitions characterised by entanglement or operator correlations. We move on to the case where the observer selects the information, introducing a novel continuous stochastic Schrodinger equation for partial post-selected (PPS) monitoring. We find that for a free fermion model, the degree of PPS introduces a new phase separation, with the phases of the post-selected dynamics remaining robust to a finite degree of PPS. Finally, we take advantage of the analytical tractability of non-Hermitian models to address the effect of initial conditions in a fully post-selected monitored free-fermionic model. The results in this thesis introduce new findings in MiPTs, along with new methods and techniques to overcome the hurdles in the field, both in the theoretical modelling and toward viable experimental protocols.
