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Equation 46 · The Entry a Relabeling Cannot Write

What does this equation mean?

DΛ=Hf−Hf=0D_\Lambda = H_f - H_f = 0

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Inputs and operationsH_f - H_f = 0
Result or conditionD_Lambda
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.

Read it piece by piece

DΛD_\Lambda

Symbol D_Lambda

DLD_Lambda is part of the quantity the equation computes from the expression on the right.

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HfH_f

Symbol H_f

HfH_f is one of the signed contributions combined to compute 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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subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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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

Read it with the definitions, units, and assumptions supplied by the article.

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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