Conjugate measurement-induced evolution towards maximally mixed states in quadratic quantum systems

In an earlier work, we showed that, in the interaction-dominant setting, coupling the conjugate variables of a subsystem to distinct environment modes drives the subsystem to a maximally mixed state. Here, we extend the study to the full quadratic dynamics of a single-mode subsystem coupled to a two-mode environment, and identify the phase-space mechanism responsible for the asymptotic behavior. By tracking the exact reduced Weyl characteristic and Wigner functions, we demonstrate that they lose their dependence on local phase-space coordinates on every fixed compact region. In free-particle model, this happens for all values of coupling, whereas in the harmonic oscillator model, it happens above a critical value of the coupling. Under a finite-volume and momentum regularization, this behavior corresponds to the reduced subsystem approaching a maximally mixed state.