Note that the recent work of the eCMM is developed based on manually adding an extra total population constraint (as described in Eq. 57) on the MMST mapping oscillator phase space. Historically, it was realized that a mapping from the quantum Schrödinger's equation to 2N classical phase space Hamilton's EOMs is incorrect, unless a total population constraints is applied. In the SU (N ) framework, on the other hand, the total population constraint is naturally satisfied through the S-W transform, without the necessity to introduce it as an additional constraint. Nevertheless, the eCMM approach derives equivalent kernel as the S-W kernel in Eq. 52 (hence also equivalent estimators in Eq. 54 and Eq. 55) from a seemingly different procedure that applies population constraint on the MMST mapping oscillator phase space. The mathematical reason behind the equivalence of two kernels is: using the additional population constraint, the 2N -dimensional MMST phase space of {q n , p n } is reduced to a complex projective (CP) space, mathematically denoted as CP(N -1), which is in fact a subspace of the parameterized manifold of SU (N ).