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Equation 43 · The Entry a Relabeling Cannot Write

What does this equation mean?

Λ†(Oc)=W†OcW\Lambda^\dagger(O_c) = W^\dagger O_c W

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Inputs and operationsW^dagger O_c W
Result or conditionLambda^dagger(O_c)
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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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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OcO_c

Symbol O_c

OcO_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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W†W^\dagger

Symbol W^dagger

WdW^dagger is an input to the expression that computes the quantity on the left.

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WW

Symbol W

the dual of conjugation by.

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=

=

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

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subscript

subscript

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.

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How to interpret it

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What the article says around this equation

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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