  The dynamics of a quantum-many body system subject to measurements is
naturally described by an ensemble of quantum trajectories, which can feature
measurement-induced phase transitions (MIPTs). This phenomenon cannot be
revealed through ensemble-averaged observables, but it requires the ability to
discriminate each trajectory separately, making its experimental observation
extremely challenging. We explore the fate of MIPTs under an observer's reduced
ability to discriminate each measurement outcome. This introduces uncertainty
in the state of the system, causing observables to probe a restricted subset of
trajectories rather than a single one. By introducing an exactly-solvable
Liouvillian model, we examine how long-time spatial correlations are influenced
by varying degrees of trajectory averaging. We compute exactly the correlation
matrix, Liouvillian gap, and entanglement negativity to demonstrate that
averaging over multiple realizations introduces an effective finite
lengthscale, beyond which long-range correlations are suppressed. This suggests
that partial averaging over trajectories conceals the critical features of
individual realizations, thereby blurring away the signatures of distinct
measurement-induced phases.
