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Λ(ρ)=TrB[ρ]\Lambda(\rho) = \mathrm{Tr}_B[\rho]

Why this formula appears here

A coarse description reports less. It is built from a completely positive, trace-preserving map Λ\Lambda , a quantum channel sending fine states to coarse states, together with a coarse Hilbert space Hc\mathcal H_c and a Hermitian coarse Hamiltonian HcH_c on it. The simplest case is the partial trace over the bath, Λ(ρ)\Lambda(\rho) = TrB\mathrm{Tr}_B[ρ\rho] with Hc\mathcal H_c = HS\mathcal H_S ; nothing below depends on that particular choice, and a passive relabeling built from no discarding at all, treated next, is a channel too.

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ρ\rho

Symbol ρ

ρ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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Λ(ρ)=TrB[ρ]\Lambda(\rho) = \mathrm{Tr}_B[\rho]

Equation 10 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

A coarse description reports less. It is built from a completely positive, trace-preserving map Λ\Lambda , a quantum channel sending fine states to coarse states, together with a coarse Hilbert space Hc\mathcal H_c and a Hermitian coarse Hamiltonian HcH_c on it. The simplest case is the partial trace over the bath, Λ(ρ)\Lambda(\rho) = TrB\mathrm{Tr}_B[ρ\rho] with Hc\mathcal H_c = HS\mathcal H_S ; nothing below depends on that particular choice, and a passive relabeling built from no discarding at all, treated next, is a channel too.

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Λ(ρ)=TrB[ρ]\Lambda(\rho) = \mathrm{Tr}_B[\rho]

Equation 57 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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

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