The Liouvillians acting on [B 0 ] w (Eq. 94) and on [B i ] w (Eq. 97) are different, which is the feature of the spin mapping formalism. We emphasize that these equations are in principle exact for linear conical intersection problem that has quadratic potential, such as spin-boson problem. This is because under such conditions, H 0 is a function up to R 2 and H k is a linear function of R, and thus the truncation in Eq. 93 becomes exact because those higher order terms of Λ will not act on H 0 or H k in Eq. 83.