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Equation 10 · Part 2 · The Entry a Relabeling Cannot Write

Symbol ρ

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

What this part means

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

Its job in the formula

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

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

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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