Equation 8 · How Fast Can a Horizon Learn Which Path You Took?

What does this equation mean?

D(T)=\frac12 \left\| |h_0\rangle\langle h_0|- |h_1\rangle\langle h_1| \right\|_1.

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Result or conditionD(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.

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D

Symbol D

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

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T

Symbol T

T 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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h_0

Symbol h_0

$h_0$ is one of the signed contributions combined to compute the quantity on the left.

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h_1

Symbol h_1

$h_1$ 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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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

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What the article says around this equation

Assume for the moment that | $h_0$ $\rangle$ and | $h_1$ $\rangle$ are pure, normalized, and prepared with equal prior probability. Define their trace distinguishability $D(T)=\frac12 \left\| |h_0\rangle\langle h_0|- |h_1\rangle\langle h_1| \right\|_1$ . For two pure states,

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