Note that if the finiteness restriction on the bath is relaxed, then the strict inequalities of (ii) and (iii) of Theorem 3 will be a non-strict inequality. The construction of the realization above can also be adapted to the special case of J ( βQ ) = 0. In this case, all subsystems of B are in the same σ$_{1}$ state and Z$_{B}$ is necessarily infinite. Unfortunately, the correct energy change of B cannot be calculated directly in such a state. At first sight, it appears that the isometry changes the state of only the subsystem X$_{1}$ of B. From here, one may try to compute the change in energy of B as Δ E$_{B}$ = tr( ρ − σ$_{1}$ ) H$_{X}$$_{1}$ , which can easily be seen to be incorrect. There is a subtle energy contribution of the shift of information in an infinite chain and that contribution can only be computed correctly as in the proof above by using a bath with finite Z$_{B}$ . This section contains the proof of the sufficiency of the conditions in Theorem 2. A separate proof has to be given for the two conditions. For the special case of (ii), a bath with a finitely many levels is sufficient while for (i), B must necessarily have infinite levels. The notation below follow the convention used in the proof of Theorem 3. Sufficiency of (ii) : Let Q be an operator that satisfies the conditions in (ii). Since σ$_{βQ}$ is isospectral with ρ$_{0}$ , there is a unitary operator W on A such that For constructing the desired realization, the bath B will be chosen to be a system identical with A and its equilibrium state will be chosen to be ρ$_{0}$ . The realization is obtained by first transforming the device by W and then swapping the states of A and B. In other words, the realization isometry is U$_{AB}$ = S ( W ⊗ 1$_{B}$ ) where the swap operator is defined to satisfy S | α 〉$_{A}$ ⊗ | α $^{′}$〉$_{B}$ = | α $^{′}$〉$_{A}$ ⊗ | α 〉$_{B}$ for all | α 〉 and | α $^{′}$〉 . If the initial state of the device is ρ , then the initial and final states of AB are 27 βQ = − W $^{†}$(ln ρ$_{0}$ + S ( ρ$_{0}$ ) 1$_{A}$ ) W . (B1) ρ$_{AB}$ = ( ρ )$_{A}$ ⊗ ρ$_{0}$ , (B2) ρ ′ AB = ( ρ$_{0}$ )$_{A}$ ⊗ ( W ρW $^{†}$)$_{B}$ . (B3) Appendix B: Sufficiency Proof of Theorem 2