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Equation 96 · Part 4 · The Bit Comes Back Before the Bearing

Symbol hat J

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,
J^\hat J

What this part means

hat J is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

hat J is one of the signed contributions combined to compute the quantity on the left.

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 variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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