Equation 42 · A Reference Frame Becomes Classical by Publishing Its Orientation

What does this equation mean?

p_{\epsilon}(F)= \int_G dg\int_{d(\hat g,g)\leq\epsilon} \operatorname{Tr}\!\left[\rho_F(g)M_F(d\hat g)\right].

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsint_G dgint_d(hat g,g) ≤ epsilon Tr[rho_F(g)M_F(dhat g)]

Result or conditionp_epsilon(F)

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states a bound: one expression must stay on the indicated side of the other 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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p_{\epsilon}

Symbol p_epsilon

$p_e$ psilon is part of the quantity the equation computes from the expression on the right.

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F

Symbol F

F 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

Symbol G

G appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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d

Symbol d

d appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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g

Symbol g

g appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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\hat g

Symbol hat g

hat g appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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\epsilon

Symbol epsilon

epsilon appears in the bound of this integral. The bound states where the repeated operation starts, ends, or which values it includes.

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\rho_F

Symbol rho_F

rh $o_F$ is an input to the expression that computes the quantity on the left.

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M_F

Symbol M_F

$M_F$ 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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≤

Less than or equal to.

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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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G

Starting index or lower bound: G

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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d(\hat g,g)\leq\epsilon

Starting index or lower bound: d(hat g,g) ≤ epsilon

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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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

For a Haar-uniform prior, define the fragment success probability $p_{\epsilon}(F)= \int_G dg\int_{d(\hat g,g)\leq\epsilon} \operatorname{Tr}\!\left[\rho_F(g)M_F(d\hat g)\right]$ . This is again a definition. The Haar measure dg is normalized to one. A frequentist worst-case version could replace the average over g with an infimum; the two must not be mixed after seeing results.

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Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

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