LEP has been discussed in several works[1, 3, 12–14], especially in the context of classical information processing. However, for the purpose of treating the equality case, the following straightforward derivation is included in here. It can be deduced simply from the following wonderful identity for deviations from canonical thermal equilibrium states where denotes the relative entropy function. Only two inequalities will be used for converting this identity to the relation in Eq. (14). The first one is the nonnegativity of the relative entropy: S ( σ || ρ ) ≥ 0 with equality holding if and only if σ = ρ . The second inequality is the subadditivity of the von Neumann entropy for the final state, i.e., S ( ρ $^{′}$$_{AB}$) ≤ S ( ρ $^{′}$$_{A}$) + S ( ρ $^{′}$$_{B}$) where the equality holds if and only if the state is in product form ρ ′ AB = ρ $^{′}$$_{A}$⊗ ρ $^{′}$$_{B}$. Finally, using the fact that the initial and final states are related by an isometry we conclude that ρ ′ AB and ρ$_{AB}$ are isospectral. In particular, S ( ρ $^{′}$$_{AB}$) = S ( ρ$_{AB}$ ) = S ( ρ$_{A}$ ) + S ( ρ$_{B}$ ) and hence which is the desired inequality. It is easy to see that both of the inequalities become equalities if and only if ρ ′ AB = E ( ρ$_{A}$ ) ⊗ ρ$_{B}$ . □ Note that the equality can be satisfied only in some exceptional cases. When that hap- pens, we have ρ ′ AB = E ( ρ$_{A}$ ) ⊗ ρ$_{B}$ and therefore the state of the bath is unchanged, i.e., ρ ′ B = ρ$_{B}$ . Consequently, the average heat dumped to bath is tr ρ$_{A}$Q = 0. Moreover, since the initial and final density matrices of AB are isospectral, it follows that ρ$_{A}$ and E ( ρ$_{A}$ ) must also be isospectral. It is of some interest to investigate the issue of whether the bound in (14) can be improved by considering the device A to be part of a larger system AX where X is not involved in the realization. In other words, the inequality is also valid for any X and any initial state ρ$_{AX}$ . Can this generalized inequality be tighter than Eq. (14)? The answer is negative. To show this, introduce another system Y such that AXY is a purification of ρ$_{AX}$ . The isometry of the realization is applied on AB and 11 tr ρ$_{A}$Q ≥ k$_{B}$T ( S ( ρ$_{AX}$ ) − S (( E ⊗ I )( ρ$_{AX}$ ))) (17) Δ E$_{B}$ ≥ k$_{B}$T ( S ( ρ $^{′}$$_{B}$) − S ( ρ$_{B}$ )) ≥ k$_{B}$T ( S ( ρ$_{A}$ ) − S ( ρ $^{′}$$_{A}$)) , S ( σ || ρ ) = tr σ (ln σ − ln ρ ) (16) Δ E$_{B}$ = k$_{B}$T ( S ( ρ $^{′}$$_{B}$| | ρ$_{B}$ ) + S ( ρ $^{′}$$_{B}$) − S ( ρ$_{B}$ )) (15)