Equation 14 · The Entry a Relabeling Cannot Write

What does this equation mean?

\mathrm{Tr}\big[\Lambda(\rho)\,O_c\big] = \mathrm{Tr}\big[\rho\,\Lambda^\dagger(O_c)\big].

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Inputs and operationsTrbig[ρLambda^dagger(O_c)big]

Result or conditionTrbig[Lambda(ρ)O_cbig]

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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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\Lambda

Symbol Lambda

completely positive and every $\Lambda(\rho)$ with $\rho$ $\ge$ 0 is itself a valid state.

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\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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O_c

Symbol O_c

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

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\Lambda^\dagger

Symbol Lambda^dagger

the define.

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=

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

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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

Every channel has a dual. Define $\Lambda^\dagger$ : $\mathcal B(\mathcal H_c)$ $\to$ $\mathcal B(\mathcal H_f)$ by the pairing that must hold for every fine state and every coarse observable $O_c$ , $\mathrm{Tr}\big[\Lambda(\rho)\,O_c\big] = \mathrm{Tr}\big[\rho\,\Lambda^\dagger(O_c)\big]$ . $\Lambda^\dagger$ is positive because $\Lambda$ is completely positive and every $\Lambda(\rho)$ with $\rho$ $\ge$ 0 is itself a valid state; it is unital, $\Lambda^\dagger(I_c)$ = $I_f$ , because $\Lambda$ preserves trace for every input. This duality is the same construction Stinespring used to show that any completely positive map admits a dilation to a larger, unitary description [ 9 ] , and it lets a coarse Hamiltonian be pulled back onto the fine space without the fine space’s own structure ever…

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Every channel has a dual. Define $\Lambda^\dagger$ : $\mathcal B(\mathcal H_c)$ $\to$ $\mathcal B(\mathcal H_f)$ by the pairing that must hold for every fine state and every coarse observable $O_c$ , $\mathrm{Tr}\big[\Lambda(\rho)\,O_c\big] = \mathrm{Tr}\big[\rho\,\Lambda^\dagger(O_c)\big]$ . $\Lambda^\dagger$ is positive because $\Lambda$ is completely positive and every $\Lambda(\rho)$ with $\rho$ $\ge$ 0 is itself a valid state; it is unital, $\Lambda^\dagger(I_c)$ = $I_f$ , because $\Lambda$ preserves trace for every input. This duality is the same construction Stinespring used to show that any completely positive map admits a dilation to a larger, unitary description [ 9 ] , and it lets a coarse Hamiltonian be pulled back onto the fine space without the fine space’s own structure ever being disturbed.

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Sources cited in the surrounding passage

[9] Positive Functions on C*-Algebras ↗

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