What does this equation mean?

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 $F_Q = 4\Big[\langle\partial_g\psi|\partial_g\psi\rangle - |\langle\psi|\partial_g\psi\rangle|^2\Big] = \frac{4}{\hbar^2}\operatorname{Var}\hat J$. 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 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 $f_1$ , the achievable variance scales as 1/(N $f_1$) , 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 $N_\gamma(t)$ $\sim$ t/$\beta$ roughly independent quanta gives, as a stated phenomenological ansatz rather than a first-principles evaporation calculation,