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Equation 17 · Part 1 · The Entry a Relabeling Cannot Write

Symbol Lambda

Λ(ρ)\Lambda(\rho)
Λ\Lambda

What this part means

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

Its job in the formula

Lambda is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Where the article explains it

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.

The passage around this formula

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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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 surrounding passage

These citations provide research context; check each source for the exact claim it supports.