Mixed-State Measurement-Induced Phase Transitions in Imaginary-Time Dynamics

Mixed-state phase transitions have recently attracted growing attention as a new frontier in nonequilibrium quantum matter and quantum information. In this work, we introduce the measurement-dressed imaginary-time evolution (MDITE) as a novel framework to explore mixed-state quantum phases and decoherence-driven criticality. In this setup, alternating imaginary-time evolution and projective measurements generate a competition between coherence-restoring dynamics and decoherence-inducing events. While reminiscent of monitored unitary circuits, MDITE fundamentally differs in that the physics is encoded in decoherent mixed states rather than in quantum trajectories. We demonstrate that this interplay gives rise to a novel class of mixed-state phase transitions, using numerical simulations of the one-dimensional transverse-field Ising model and the two-dimensional columnar dimerized Heisenberg model. Notably, the observed transitions do not fall into any previously established universality classes. Furthermore, we provide a diagrammatic representation of the evolving state, which naturally enables efficient studies of MDITE with quantum Monte Carlo and other many-body numerical methods, thereby extending investigations of mixed-state phase transitions to large-scale and higher-dimensional systems. In addition, the representation provides a natural interpretation of the phase transitions in terms of cluster formation within the simulations. Our results highlight MDITE as a powerful paradigm for investigating non-unitary dynamics and the fundamental role of decoherence in many-body quantum systems.