Let { ν$_{k}$ } and { ν ′ $_{k}$} be two infinite, increasing sequences starting from 0 and converging to 1. The energy levels of X$_{k}$ and Y$_{k}$ will be taken as ϵ$_{i}$ ( ν$_{k}$ ) and ϵ ′ $_{i}$( ν ′ $_{k}$) respectively. In other words, in terms of the Hamiltonians of individual systems can be expressed as H$_{X}$$_{k}$ = h ( ν$_{k}$ ) and H$_{Y}$$_{k}$ = h $^{′}$( ν ′ $_{k}$). The bath Hamiltonian H$_{B}$ is the sum of the subsystem Hamiltonians and therefore the energy eigenvalue of the bath state | j$_{1}$j$_{2}$ · · · ; i$_{1}$i$_{2}$ · · · 〉$_{B}$ is Let σ ( ν ) = exp( − βh ( ν )) /ζ ( ν ) and σ $^{′}$( ν ) = exp( − βh $^{′}$( ν )) /ζ $^{′}$( ν ) be parameter-dependent density matrices and ζ ( ν ) and ζ $^{′}$( ν ) be the corresponding partition functions. The shorthand notations σ$_{k}$ = σ ( ν$_{k}$ ) and σ ′ k = σ $^{′}$( ν ′ $_{k}$) will be used for the states of X$_{k}$ and Y$_{k}$ respectively. The partition function of the bath is then given by Depending only on the large k behavior of the sequences { ν$_{k}$ } and { ν ′ $_{k}$} , this product may converge or diverge. (i) The zeroth level has zero energy, ϵ$_{0}$ ( ν ) = ϵ $^{′}$$_{0}$( ν ) = 0, for all ν . (ii) The energies of all excited levels diverge to + ∞ as the corresponding parameter ap- proaches to 1, i.e., (iii) The energy spectrum ϵ$_{i}$ ( ν = 0) is identical with the eigenvalues of Q up to a constant shift as ϵ$_{i}$ (0) = q$_{i}$ − q$_{0}$ for all i . (iv) The energy spectrum ϵ ′ $_{j}$( ν = 0) is such that the equilibrium density matrix is isospec- tral with ρ$_{0}$ , i.e., ϵ ′ $_{j}$(0) = k$_{B}$T ln( p$_{0}$/p$_{j}$ ). lim ν → $_{1}$ϵ$_{i}$ ( ν ) = lim ν → $_{1}$ϵ ′ $_{j}$( ν ) = ∞ (for any i, j ≥ 1) . (B11) h ( ν ) = d − 1 ∑ i =0 ϵ$_{i}$ ( ν ) | i 〉 〈 i | , (B12) h $^{′}$( ν ) = r − 1 ∑ i =0 ϵ ′ $_{i}$( ν ) | i 〉 〈 i | , (B13) E ( { j$_{k}$ } , { i$_{k}$ } ) = ∞ ∑ k =1 ϵ ′ j $_{k}$( ν ′ $_{k}$) + ϵ$_{i}$$_{k}$ ( ν$_{k}$ ) . (B14) Z$_{B}$ = ∞ ∏ k =1 ζ ( ν$_{k}$ ) ζ $^{′}$( ν ′ $_{k}$) . (B15) 30