As a result, E ( ρ ) = ρ$_{0}$ and the energy of the bath has been changed by and therefore the HTO associated with this realization is Q . This completes the proof of sufficiency of the special case (ii). Sufficiency of (i) : Let Q be a hermitian operator satisfying only the strict inequality in the statement (i). The construction of the realization is quite similar to the construction used in the proof of Theorem 3, but now the bath has more components. Let d be the dimension of the Hilbert space of the device A and let r be the matrix rank of ρ$_{0}$ .[25]. Let X be a d -level system and Y be an r -level system. The bath is taken to be composed of infinitely many copies of X and Y. The constitution of B can be expressed as B = (Y$_{1}$Y$_{2}$Y$_{3}$ · · · )(X$_{1}$X$_{2}$X$_{3}$ · · · ). Let the standard orthonormal bases of these two types of subsystems be denoted by the common notation {| i 〉} where i = 0 , 1 , . . . , d − 1 for X and i = 0 , 1 , . . . , r − 1 for Y subsystems. The following shorthand will be used for the state vectors of the bath The Hilbert space H$_{B}$ of the bath is defined as the linear span of these states where the associated sequences { j$_{k}$ } and { i$_{k}$ } contain only finitely many nonzero elements. These sequences can be considered as the digits of base- r and base- d representation of two non- negative integers m and n respectively. Namely, m = ( · · · j$_{3}$j$_{2}$j$_{1}$ )$_{r}$ and n = ( · · · i$_{3}$i$_{2}$i$_{1}$ )$_{d}$. Therefore, H$_{B}$ has a countable basis and it is a separable Hilbert space. Let {| i 〉} and {| α$_{i}$ 〉} be the orthonormal sets in the Hilbert space of the device A that appear in the spectral decompositions of Q and ρ$_{0}$ as in Eq. (29) and where the eigenvalues are ordered in the indicated way. The isometry U$_{AB}$ on the composite system is defined as follows U$_{AB}$ | ℓ 〉$_{A}$ ⊗ | j$_{1}$j$_{2}$j$_{3}$ · · · ; i$_{1}$i$_{2}$i$_{3}$ · · · 〉$_{B}$ = | α$_{j}$$_{1}$ 〉$_{A}$ ⊗ | j$_{2}$j$_{3}$j$_{4}$ · · · ; ℓi$_{1}$i$_{2}$ · · · 〉$_{B}$ , (B10) ρ$_{0}$ = r − 1 ∑ i =0 p$_{i}$ | α$_{i}$ 〉 〈 α$_{i}$ | ( p$_{0}$ ≥ p$_{1}$ ≥ · · · ≥ p$_{r}$$_{−}$$_{1}$ > 0) (B9) | j$_{1}$j$_{2}$ · · · ; i$_{1}$i$_{2}$ · · · 〉$_{B}$ = ( | j$_{1}$ 〉$_{Y}$$_{1}$ ⊗ | j$_{2}$ 〉$_{Y}$$_{2}$ ⊗ · · · ) ⊗ ( | i$_{1}$ 〉$_{X}$$_{1}$ ⊗ | i$_{2}$ 〉$_{X}$$_{2}$ ⊗ · · · ) . (B8) Δ E$_{B}$ = tr H$_{B}$ ( W ρW † − ρ$_{0}$ ) (B4) = − k$_{B}$T tr ln ρ$_{0}$ ( W ρW † − ρ$_{0}$ ) (B5) = − k$_{B}$T tr ρ ( W $^{†}$ln ρ$_{0}$W + S ( ρ$_{0}$ ) 1$_{B}$ ) (B6) = tr ρQ (B7) 28