When a quantum system is coupled to a dissipative environment an initially pure state becomes rapidly mixed as information is lost, and classical behavior invariably follows. Recently, another type of open system dynamics has been explored, when a quantum system is continuously "monitored" by an observer, making a sequence of measurements, and a pure quantum state remains pure. The resulting quantum trajectories constitute an ensemble of pure states, which can in the many-body context, have a rich entanglement structure, exhibiting measurement-induced phase transitions (MIPT) between volume law and area law entanglement, and between phases with or without symmetry breaking and/or topological order. For mixed initial density matrices, monitoring can lead to a plethora of purification transitions, and reveals underlying connections with quantum encoding. While the MIPT occurs generically in a number of different models, its verification can be challenging even on an error-corrected quantum computer, due to the so-called post-selection problem. I will briefly describe recent proposals to access experimentally measurement-induced phase transitions (MIPT) employing \decoding", including the prospects of employing the linear cross-entropy, without requiring any post- selection of quantum trajectories.