In other words, in terms of the Hamiltonians of individual subsystems can be expressed as H$_{X}$$_{k}$ = h ( ν$_{k}$ ). The bath Hamiltonian H$_{B}$ is the sum of the subsystem Hamiltonians and therefore the energy eigen- value of the bath state | n 〉$_{B}$ = | i$_{1}$, i$_{2}$, i$_{3}$, · · · 〉$_{B}$ is Note that if { i$_{k}$ } had infinitely many nonzero entries, then the corresponding energy would be infinite. This justifies the definition of the Hilbert space H$_{B}$ in the way described above. Let σ ( ν ) = exp( − βh ( ν )) /ζ ( ν ) be the parameter-dependent density matrix and ζ ( ν ) be the corresponding partition function. The shorthand notation σ$_{k}$ = σ ( ν$_{k}$ ) will be used for the thermal equilibrium states of X$_{k}$. The partition function of the bath is then given by Note that the convergence of this product depends only on the large k behavior of the sequence { ν$_{k}$ } . By adjusting the way ν$_{k}$ converges to 1, it is possible obtain a finite Z$_{B}$ . Let us now compute the HTO. In the following expressions, the subscripts are used for indicating the subsystem a given density matrix applies to. Suppose that the initial state of the device A is ρ . The initial and final states of AB are given by the following self-explanatory expressions. The change in the average energy of the bath is given by where J = J ( βQ ) and 25 Δ = k$_{B}$T J − q$_{0}$ − tr σ$_{1}$H$_{X}$$_{1}$ + ∞ ∑ k =2 tr( σ$_{k}$$_{−}$$_{1}$ − σ$_{k}$ ) H$_{X}$$_{k}$ , (A7) Δ E$_{B}$ = tr( ρQ ) − k$_{B}$T J + Δ (A6) ρ$_{AB}$ = ( ρ )$_{A}$ ⊗ ( σ$_{1}$ )$_{X}$$_{1}$ ⊗ ( σ$_{2}$ )$_{X}$$_{2}$ ⊗ · · · , (A4) ρ ′ AB = ( | ψ$_{0}$ 〉 〈 ψ$_{0}$ | )$_{A}$ ⊗ ( ρ )$_{X}$$_{1}$ ⊗ ( σ$_{1}$ )$_{X}$$_{2}$ ⊗ · · · . (A5) Z$_{B}$ = ∞ ∏ k =1 ζ ( ν$_{k}$ ) . (A3) E$_{n}$ = ∞ ∑ k =1 ϵ$_{i}$$_{k}$ ( ν$_{k}$ ) . (A2) h ( ν ) = d − 1 ∑ i =0 ϵ$_{i}$ ( ν ) | i 〉 〈 i | , (A1)