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Equation 2 · Part 10 · How Fast Can a Horizon Learn Which Path You Took?

Numerator: |0rangle|h_0(T)rangle+ |1rangle|h_1(T)rangle

∣Ψ(T)⟩=∣0⟩∣h0(T)⟩+∣1⟩∣h1(T)⟩2.|\Psi(T)\rangle= \frac{|0\rangle|h_0(T)\rangle+ |1\rangle|h_1(T)\rangle}{\sqrt 2}.
∣0⟩∣h0(T)⟩+∣1⟩∣h1(T)⟩|0\rangle|h_0(T)\rangle+ |1\rangle|h_1(T)\rangle

What this part means

The complete quantity above the fraction bar.

Its job in the formula

|0rangle|h0(T)h_0(T)rangle+ |1rangle|h1(T)h_1(T)rangle occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

The labels zero and one denote paths, not energy eigenstates. After the system’s long-range field interacts with the inaccessible degrees of freedom, an idealized final pure state has the form ∣Ψ(T)⟩=∣0⟩∣h0(T)⟩+∣1⟩∣h1(T)⟩2|\Psi(T)\rangle= \frac{|0\rangle|h_0(T)\rangle+ |1\rangle|h_1(T)\rangle}{\sqrt 2}. Tracing out the field multiplies the off-diagonal element of the path density matrix by the conditional-state overlap

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the article section

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