In the following expressions, the subscripts are used for indicating the subsystem a given density matrix specifies to. Suppose that the initial state of the device A is ρ . The initial and final density matrices of the combined system are given by the following self-explanatory expressions. It is obvious that the reduced quantum operation on the device is E ( ρ ) = ρ$_{0}$ , i.e., the desired complete erasure. The change in the average energy of the bath is given by where J = J ( βQ ) and which is a constant independent of the device state ρ . When Eq. (15) is used, Δ can be expressed as which shows that Δ is a strictly positive number. It will be argued below that the sequences of subsystem parameters { ν$_{k}$ } and { ν ′ $_{k}$} can be selected in such a way that Δ can be adjusted to be equal to any positive value and this can be achieved with a finite Z$_{B}$ . As in the proof of Theorem 3, this will be done by showing that Δ can be made arbitrarily large and arbitrarily small. (1) First, if the parameter ν$_{2}$ of X$_{2}$ tends to 1, the final energy of X$_{2}$ approaches to infinity. In this limit Δ diverges to ∞ . (ii) In the limit where all increments ν$_{k}$$_{+1}$ − ν$_{k}$ and ν ′ k +1 − ν ′ k tend to 0, the series in Eq. (B17) approach to integrals which can readily be evaluated as in Eq. (A9) and 31 ∞ ∑ k =1 tr( σ ′ k +1 − σ ′ $_{k}$) H$_{Y}$$_{k}$ −→ − k$_{B}$T S ( σ $^{′}$(0)) . (B19) Δ = k$_{B}$T ∞ ∑ k $_{=1}$( S ( σ$_{k}$ || σ$_{k}$$_{+1}$ ) + S ( σ ′ k $_{+1}$| | σ ′ $_{k}$) ) , (B18) Δ = k$_{B}$T ( J + S ( ρ$_{0}$ )) − tr σ$_{1}$Q + ∞ ∑ k =2 tr( σ$_{k}$$_{−}$$_{1}$ − σ$_{k}$ ) H$_{X}$$_{k}$ + ∞ ∑ k =1 tr( σ ′ k +1 − σ ′ $_{k}$) H$_{Y}$$_{k}$ , (B17) Δ E$_{B}$ = tr( ρQ ) − k$_{B}$T ( J + S ( ρ$_{0}$ )) + Δ (B16) ρ$_{AB}$ = · · · ( θ$_{2}$ )$_{Y}$$_{2}$ ⊗ ( θ$_{1}$ )$_{Y}$$_{1}$ ⊗ ( ρ )$_{A}$ ⊗ ( σ$_{1}$ )$_{X}$$_{1}$ ⊗ ( σ$_{2}$ )$_{X}$$_{2}$ · · · ρ ′ AB = · · · ( θ$_{3}$ )$_{Y}$$_{2}$ ⊗ ( θ$_{2}$ )$_{Y}$$_{1}$ ⊗ ( ρ$_{0}$ )$_{A}$ ⊗ ( ρ )$_{X}$$_{1}$ ⊗ ( σ$_{1}$ )$_{X}$$_{2}$ · · ·