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Equation 44 · Part 7 · The Entry a Relabeling Cannot Write

subscript

Λ†(Hc)=W†(WHfW†)W=(W†W)Hf(W†W)=Hf,\Lambda^\dagger(H_c) = W^\dagger\big(W H_f W^\dagger\big) W = (W^\dagger W) H_f (W^\dagger W) = H_f,
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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, Λ†(Hc)=W†(WHfW†)W=(W†W)Hf(W†W)=Hf\Lambda^\dagger(H_c) = W^\dagger\big(W H_f W^\dagger\big) W = (W^\dagger W) H_f (W^\dagger W) = H_f. using only W†W^\dagger W = I . So DΛD_\Lambda = HfH_f - HfH_f = 0 , identically, as an operator, for every unitary W and every HfH_f . This is not a limit, an approximation, or a property of some special state; it is an algebraic identity that holds before any state is chosen at all.

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

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