addition

The passage around this formula

Return to the genuine coarse-graining set aside earlier: $\Lambda(\rho)$ = $\mathrm{Tr}_B$[$\rho$] , $\mathcal H_c$ = $\mathcal H_S$ , $H_c$ = $H_S$ . The dual embeds a system observable back into the full space without touching the bath, $\Lambda^\dagger(H_S)$ = $H_S$ $\otimes$ $I_B$ , so $D_\Lambda = H_f - H_S \otimes I_B = I_S \otimes H_B + H_{SB}$. an exact operator identity, not an approximation valid in some limit: the ledger for a system-only description of a system-plus-bath is precisely the bath’s own energy plus the coupling energy between the two, nothing more and nothing less. $\Delta$ $E_\Lambda(\rho)$ = $\mathrm{Tr}$[$\rho(I_S \otimes H_B + H_{SB})$] for any joint state $\rho$ .

Read this part in the article →