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Equation 96 · Part 9 · The Bit Comes Back Before the Bearing

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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,
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What this part means

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.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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