Semiclassical approximations to quantum mechanics can include quantum coherence effects in dynamical calculations based on classical mechanics. The Herman-Kluk (HK) semiclassical propagator has been demonstrated to reproduce but does not provide a practical numerical route to calculations for multiple degrees of freedom. In an HK calculation of a response function, quantum coherence effects enter through interference between pairs of classical trajectories. We have previously quantum effects in nonlinear vibrational response functions of anharmonic oscillators elucidated the mechanism by which the HK approximation reproduces quantum effects in response functions in the regime of quasiperiodic dynamics. We have applied this understanding to significantly simplify the semiclassical calculation of response functions in this dynamical regime. The phase space difference between trajectories is treated perturbatively in anharmonicity, allowing integration over these differences to be performed analytically and leaving integration over mean trajectories to be performed numerically. This mean-trajectory (MT) approximation has been applied to linear and nonlinear vibrational response functions for isolated and coupled anharmonic motions. Here, we derive an MT approximation for the Liouville space time evolution operator or superoperator that propagates the density operator. This analysis provides a form of the MT approximation that is readily applicable to other dynamical quantities besides response functions and clarifies the connection between semiclassical quantization of propagators for the wave function and for the density operator.