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The confluence of unitary dynamics and non-unitary measurements gives rise tointriguing and relevant phenomena, generally referred to as measurement-inducedphase transitions. These transitions have been observed in quantum systemscomposed of trapped ions and superconducting circuits. However, theirexperimental realization demands substantial resources, primarily owing to theclassical tracking of measurement outcomes, known as post-selection oftrajectories. In this work, we first describe the statistical properties of aninteracting transmon array which is repeatedly measured and predict thebehavior of relevant quantities in the area-law phase using a combination ofthe replica method and non-Hermitian perturbation theory. We show that by usingpredetermined measurements that force the system to be locally in a certainstate after performing a measurement we can make use of the distribution of thenumber of bosons measured at a single site as an indirect diagnostic for thephase transition in the entanglement of individual trajectories. Interestingly,a phase transition is obtained without considering an absorbing state. Thisimplies that the predetermined measurement approach might be a viableexperimental option to determine the phase of the system without requiringpost-selection. We also show numerically that a transmon array, modeled by anattractive Bose-Hubbard model, in which local measurements of the number ofbosons are probabilistically interleaved, exhibits a phase transition in theentanglement entropy properties of the ensemble of trajectories in the steadystate.

Phase transitions induced by standard and predetermined measurements in transmon arrays