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Equation 41 · Part 5 · The Entry a Relabeling Cannot Write

=

Tr[WρW†Oc]=Tr[ρW†OcW]\mathrm{Tr}[W\rho W^\dagger O_c] = \mathrm{Tr}[\rho W^\dagger O_c W]
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What this part means

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

The dual of conjugation by W is conjugation by W†W^\dagger : from the defining pairing, Tr\mathrm{Tr}[Wρ\rho W†W^\dagger OcO_c] = Tr\mathrm{Tr}[ρ\rho W†W^\dagger OcO_c W] holds for every ρ\rho , so Λ†(Oc)\Lambda^\dagger(O_c) = W†W^\dagger OcO_c W . Substituting,

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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