In CSDM, tCSDM, and κCSDM, we propagate two density matrices; the true density matrix is propagated by equations 22 and 14, and another density matrix, called the coherent density matrix, is propagated coherently (i.e., by eqs 12 and 14) and is used to control the switching of the pointer state. Although the coherent density matrix is in general different from true density along the trajectory, it is re-initialized to true density for every complete passage of a strong interaction region.