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Equation 10 · How Fast Can a Horizon Learn Which Path You Took?

What does this equation mean?

pe(T)=1−D(T)2=1−1−V(T)22.p_{\rm e}(T)=\frac{1-D(T)}{2} =\frac{1-\sqrt{1-V(T)^2}}{2}.

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.

Start with1-D(T)
Divide by2
This relates top_rm e(T)
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

pep_{\rm e}

Symbol p_rm e

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

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TT

Symbol T

T occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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DD

Symbol D

D occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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VV

Symbol V

V occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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=

=

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

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fraction

fraction

Divide the expression above the line by the one below it.

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√

√

Take a square root.

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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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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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1−D(T)1-D(T)

Numerator: 1-D(T)

The complete quantity above the fraction bar.

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22

Denominator: 2

The complete quantity below the fraction bar; it must be nonzero for this division.

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1−1−V(T)21-\sqrt{1-V(T)^2}

Numerator: 1-sqrt1-V(T)^2

The complete quantity above the fraction bar.

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22

Denominator: 2

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

Helstrom’s binary decision theory supplies the minimum average probability of guessing the wrong path: pe(T)=1−D(T)2=1−1−V(T)22p_{\rm e}(T)=\frac{1-D(T)}{2} =\frac{1-\sqrt{1-V(T)^2}}{2}. For the balanced pure-state problem, the minimum-error measurement induces a symmetric binary channel. Its recovered decision information is

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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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