Additionally, in the Ehrenfest+R approach, it was numerically found that adding random phase only to c 0,ξ (t + dt) gave the most accurate results; 11 however, these choices lacked a rigorous theoretical reason for why this should be the case and it was speculated that this is because spontaneous emission from states |1⟩ to |0⟩ should not affect the coherence of state |1⟩ with other states {|j⟩}. In contrast, the analysis in this paper shows that the mathematical reason for this choice of random phase lies in the fact that the excited state |1⟩ decoheres with every other state of the quantum subsystem while state |0⟩ only decoheres with state |1⟩, when L = |0⟩⟨1|. This fact is derived from applying the Lindbladian L L (with L = |0⟩⟨1|) to the entire reduced density matrix of K states instead of only the reduced density matrix of |0⟩ and |1⟩. While state |1⟩ decoheres with every other state, the decoherence with states {|j⟩} is entirely captured by the reduction of the magnitude of c 1,ξ (t + dt) by e -Γdt/2 ; thus, the phase of state |1⟩ does not need to provide any additional decoherence. In contrast, state |0⟩ does not decohere with states {|j⟩}, but the increase of the magnitude of c 0,ξ (t + dt) causes an increase in coherence with states {|j⟩}. The role of the random phase applied to state |0⟩, aside from adjusting the coherence with state |1⟩, is to add decoherence to cancel out the increase in coherence with states {|j⟩} due to the increase of the magnitude of c 0,ξ (t + dt).