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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. 

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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 

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which is a constant independent of the device state ρ . 

When Eq. (15) is used, Δ can be expressed as 

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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 ∞ 

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