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Equation 45 · The Entry a Relabeling Cannot Write

What does this equation mean?

W†W=IW^\dagger W = I

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Inputs and operationsI
Result or conditionW^dagger W
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

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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W†W^\dagger

Symbol W^dagger

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

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WW

Symbol W

the dual of conjugation by.

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II

Symbol I

I is an input to the expression that computes the quantity on the left.

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=

=

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

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How to interpret it

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What the article says around this equation

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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