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Equation 96 · Part 11 · The Bit Comes Back Before the Bearing

Denominator: hbar^2

FQ=4[⟨∂gψ∣∂gψ⟩−∣⟨ψ∣∂gψ⟩∣2]=4ℏ2Var⁡J^,F_Q = 4\Big[\langle\partial_g\psi|\partial_g\psi\rangle - |\langle\psi|\partial_g\psi\rangle|^2\Big] = \frac{4}{\hbar^2}\operatorname{Var}\hat J,
ℏ2\hbar^2

What this part means

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

Its job in the formula

hbar2r^2 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

The compass’s Fisher information does not have an analogous reason to jump. For a pure-state family generated by a Hermitian charge, the exact quantum Fisher information is FQ=4[⟨∂gψ∣∂gψ⟩−∣⟨ψ∣∂gψ⟩∣2]=4ℏ2Var⁡J^F_Q = 4\Big[\langle\partial_g\psi|\partial_g\psi\rangle - |\langle\psi|\partial_g\psi\rangle|^2\Big] = \frac{4}{\hbar^2}\operatorname{Var}\hat J. a standard identity in quantum metrology [ 12 ] . Before scrambling, this is fixed by how the compass itself was built — a large, well-prepared gyroscope has a large charge variance and a small intrinsic uncertainty on its own heading. After scrambling, that fixed total has to be reconstructed piecemeal from radiation, and each individually emitted quantum, to the extent it is only weakly and independently correlated with g once the hole’s own state is traced over, contributes an addition to the total…

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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 surrounding passage

These citations provide research context; check each source for the exact claim it supports.