Measurement-induced phase transitions and bistability in quantum–classical hybrids

Abstract This paper introduces a novel mathematical framework for analyzing bifurcations in quantum–classical hybrid systems, bridging quantum information science with classical nonlinear dynamics. Our approach integrates noncommutative bifurcation theory, quantum center manifold reduction, and measurement-induced transition analysis to provide rigorous mathematical results supported by detailed numerical examples. The proposed model captures bidirectional quantum–classical interactions through coupled differential equations, where quantum measurement backaction and entanglement play fundamental roles in modifying bifurcation behavior. We demonstrate significant advances in understanding quantum–classical interplay, with promising applications in quantum sensing, control, and synchronization. Our method is general and applicable to various nonlinear systems as verified by our numerical examples.