Nonstabilizerness in the unitary and monitored quantum dynamics of XXZ-staggered and SYK models

We consider the quantum-state-diffusion dynamics of the XXZ-staggered spin chain, also focusing on its noninteracting XX-staggered limit, and of the Sachdev-Ye-Kitaev (SYK) model. We describe the process through quantum trajectories and evaluate the nonstabilizerness (also known as ``magic'') along the trajectories, quantified through the stabilizer R\'enyi entropy (SRE). In the absence of measurements, we find that the SYK model is the only one in which the time-averaged SRE saturates the random state bound and has a scaling with the system size that is well described by the theoretical prediction for quantum chaotic systems. In the presence of measurements, we numerically find that the steady-state SRE versus the coupling strength to the environment is well fitted by a generalized Lorentzian function. The scaling of the fitting parameters with the system size suggests that the steady-state} SRE linearly increases with the system size in all the considered cases, and displays no measurement-induced quantum transition, as confirmed by the curves of the steady-state SRE versus the system size.