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Equation 99 · The Bit Comes Back Before the Bearing

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f1f_1

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the per-probe fisher information. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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f1f_1

Symbol f_1

the per-probe fisher information.

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subscript

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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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 that adds like an independent sample rather than like a decoded codeword. This is the standard-quantum-limit regime of parameter estimation: for N independent, identically prepared probes each…
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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 that adds like an independent sample rather than like a decoded codeword. This is the standard-quantum-limit regime of parameter estimation: for N independent, identically prepared probes each with per-probe Fisher information f1f_1 , the achievable variance scales as 1/(N f1f_1) , in contrast to the quadratically better Heisenberg scaling available only when probes are used coherently together [ 13 ] . Modeling the collected radiation up to time t as contributing Nγ(t)N_\gamma(t) ∼\sim t/β\beta roughly independent quanta gives, as a stated phenomenological ansatz rather than a first-principles evaporation calculation,

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