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Published equation contexts

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

Why this formula appears here

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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Λ†\Lambda^\dagger

Symbol Lambda^dagger

Lambdada^dagger is part of the quantity the equation computes from the expression on the right.

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HcH_c

Symbol H_c

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

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Published contexts (1)

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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,

Equation 44 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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.

Meanings in this article

  • WW: the dual of conjugation by.
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