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Equation 10 · The Entry a Relabeling Cannot Write

What does this equation mean?

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

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Inputs and operationsTr_B[ρ]
Result or conditionLambda(ρ)
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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Λ\Lambda

Symbol Lambda

the trace-preserving map.

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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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BB

Symbol B

B is an input to the expression that computes the quantity on the left.

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=

=

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

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

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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