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Non-equilibrium dynamics of many-body quantum systems under the effect ofmeasurement protocols is attracting an increasing amount of attention. It hasbeen recently revealed that measurements may induce different non-equilibriumregimes and an abrupt change in the scaling-law of the bipartite entanglemententropy. However, our understanding of how these regimes appear, how theyaffect the statistics of local quantities and, finally whether they survive inthe thermodynamic limit, is much less established. Here we investigatemeasurement-induced phase transitions in the Quantum Ising chain coupled to amonitoring environment. In particular we show that local projectivemeasurements induce a quantitative modification of the out-of-equilibriumprobability distribution function of the local magnetization. Starting from aGHZ state, the relaxation of the paramagnetic and the ferromagnetic order isanalysed. In particular we describe how the probability distribution of theformer shows different behaviour in the area-law and volume-law regimes.

Full counting statistics as probe of measurement-induced transitions in the quantum Ising chain