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Equation 10 · Part 1 · The Entry a Relabeling Cannot Write

Symbol Lambda

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

What this part means

the trace-preserving map.

Its job in the formula

Lambda is part of the quantity the equation computes from the expression on the right.

Where the article explains it

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 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 function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the article section

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