Note that because the quantum Liouvillian is derived based on B(t) s = e L(rs)t B s , thus L will have the same S-W radius r s as B. From our rigorous theoretical formalism we justify that the index used to define {q n , p n } has to match the index of the Hamiltonian as was numerically found previously, because it is the choice that leads to an initial distribution equivalent to the one obtained from dΩ (as no index is involved in dΩ only one of the two choices of radius leads to the same distribution). Thus, one needs to fix r s for the quantum Liouvillian, such that the index of the Liouvillian coincides with the index used to define {q n , p n } in Eq. 46. During this process, the index of B s is also fixed, and one will have Â s with the complementing index. In the Spin-LSC approach, on the other hand, L contains the same choice of r s as used in [ Â] s, and operator B has a different complementing radius r s . Same choice is also made in eCMM. It was numerically demonstrated that this choice leads to a better population dynamics for a spin-boson problem 85 (for the s = Q choice in [ Â] s). However, these results are usually less accurate compared to the choice of r s = r s = √ N + 1. Thus, we will only consider the case of r s = r s in this work, and under this special case, our choice of r s in L do agree with those in Spin-LSC.