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A coarse description reports less. It is built from a completely positive, trace-preserving map $\Lambda$ , a quantum channel sending fine states to coarse states, together with a coarse Hilbert space $\mathcal H_c$ and a Hermitian coarse Hamiltonian $H_c$ on it. The simplest case is the partial trace over the bath, $\Lambda(\rho)$ = $\mathrm{Tr}_B$[$\rho$] with $\mathcal H_c$ = $\mathcal H_S$ ; 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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