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Equation 10 · Part 4 · The Entry a Relabeling Cannot Write

=

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

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

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