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Equation 63 · A Reference Frame Becomes Classical by Publishing Its Orientation

What does this equation mean?

d(hg^,hg)=d(g^,g).d(h\hat g,hg)=d(\hat g,g).

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Inputs and operationsd(hat g,g)
Result or conditiond(hhat g,hg)
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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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dd

Symbol d

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

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hh

Symbol h

h 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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g^\hat g

Symbol hat g

hat g 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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gg

Symbol g

the covariance sends.

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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

The central mathematical test is simple to state. Apply a common frame transformation h to the physical description. Covariance sends g and g^\hat g to hg and hg^\hat g , or to the appropriate left/right action fixed by convention. A left-invariant distance satisfies d(hg^,hg)=d(g^,g)d(h\hat g,hg)=d(\hat g,g). The Haar measure is invariant, and the transformed POVM produces the correspondingly transformed outcome probabilities. Therefore the integration region defining success is merely relabeled. The value of pϵ(F)p_\epsilon(F) , and hence Rϵ,δR_{\epsilon,\delta} , should not change.

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