Equation 20 · The Entry a Relabeling Cannot Write

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\Lambda

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completely positive and every $\Lambda(\rho)$ with $\rho$ $\ge$ 0 is itself a valid state. 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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$\Lambda^\dagger$ 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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[9] Positive Functions on C*-Algebras ↗

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