Universal Stochastic Equations of Monitored Quantum Dynamics

We investigate the monitored quantum dynamics of Gaussian mixed states and derive the universal Fokker-Planck equations that govern the stochastic time evolution of entire density-matrix spectra, obtaining their exact solutions. From these equations, we reveal an even-odd effect in purification dynamics: whereas entropy exhibits exponential decay for an even number N of complex fermions, algebraic decay with divergent purification time occurs for odd N as a manifestation of dynamical criticality. Additionally, we identify the universal fluctuations of entropy in the chaotic regime, serving as a nonunitary counterpart of the universal conductance fluctuations in mesoscopic electronic transport phenomena. Furthermore, we elucidate and classify the universality classes of nonunitary quantum dynamics based on fundamental symmetry. We also validate the universality of these analytical results through extensive numerical simulations across different types of models.