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ρ≥0\rho \ge 0

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Λ†\Lambda^\dagger is positive because Λ\Lambda is completely positive and every Λ(ρ)\Lambda(\rho) with ρ\rho ≥\ge 0 is itself a valid state; it is unital, Λ†(Ic)\Lambda^\dagger(I_c) = IfI_f , because Λ\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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ρ\rho

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ρ≥0\rho \ge 0

Equation 18 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

Λ†\Lambda^\dagger is positive because Λ\Lambda is completely positive and every Λ(ρ)\Lambda(\rho) with ρ\rho ≥\ge 0 is itself a valid state; it is unital, Λ†(Ic)\Lambda^\dagger(I_c) = IfI_f , because Λ\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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