← All parts of this equation

Equation 14 · Part 1 · The Entry a Relabeling Cannot Write

Symbol Lambda

Tr[Λ(ρ) Oc]=Tr[ρ Λ†(Oc)].\mathrm{Tr}\big[\Lambda(\rho)\,O_c\big] = \mathrm{Tr}\big[\rho\,\Lambda^\dagger(O_c)\big].
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 part of the quantity the equation computes from the expression on the right.

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

…Define Λ†\Lambda^\dagger: B(Hc)\mathcal B(\mathcal H_c) →\to B(Hf)\mathcal B(\mathcal H_f) by the pairing that must hold for every fine state and every coarse observable OcO_c , Tr[Λ(ρ) Oc]=Tr[ρ Λ†(Oc)]\mathrm{Tr}\big[\Lambda(\rho)\,O_c\big] = \mathrm{Tr}\big[\rho\,\Lambda^\dagger(O_c)\big]. 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…

Read this part in the article →

Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

Open the illustrated functions: inputs become outputs guide →

See this notation across published equations →

Sources cited in the surrounding passage

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