← All parts of this equation

Equation 44 · Part 2 · The Entry a Relabeling Cannot Write

Symbol H_c

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,
HcH_c

What this part means

HcH_c is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Its job in the formula

HcH_c is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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.

Read this part in the article →

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 →

See this notation across published equations →

The article lists its research sources here.