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Equation 61 · Part 7 · The Entry a Relabeling Cannot Write

Symbol H_SB

DΛ=Hf−HS⊗IB=IS⊗HB+HSB,D_\Lambda = H_f - H_S \otimes I_B = I_S \otimes H_B + H_{SB},
HSBH_{SB}

What this part means

HSH_SB is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

HSH_SB is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

Return to the genuine coarse-graining set aside earlier: Λ(ρ)\Lambda(\rho) = TrB\mathrm{Tr}_B[ρ\rho] , Hc\mathcal H_c = HS\mathcal H_S , HcH_c = HSH_S . The dual embeds a system observable back into the full space without touching the bath, Λ†(HS)\Lambda^\dagger(H_S) = HSH_S ⊗\otimes IBI_B , so DΛ=Hf−HS⊗IB=IS⊗HB+HSBD_\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. Δ\Delta EΛ(ρ)E_\Lambda(\rho) = Tr\mathrm{Tr}[ρ(IS⊗HB+HSB)\rho(I_S \otimes H_B + H_{SB})] for any joint state ρ\rho .

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A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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