  Continuously monitored atomic spin-ensembles allow, in principle, for
real-time sensing of external magnetic fields beyond classical limits. Within
the linear-Gaussian regime, thanks to the phenomenon of measurement-induced
spin-squeezing, they attain a quantum-enhanced scaling of sensitivity both as a
function of time, $t$, and the number of atoms involved, $N$. In our work, we
rigorously study how such conclusions change when inevitable imperfections are
taken into account: in the form of collective noise, as well as stochastic
fluctuations of the field in time. We prove that even an infinitesimal amount
of noise disallows the error to be arbitrarily diminished by simply increasing
$N$, and forces it to eventually follow a classical-like behaviour in $t$.
However, we also demonstrate that, "thanks" to the presence of noise, in most
regimes the model based on a homodyne-like continuous measurement actually
achieves the ultimate sensitivity allowed by the decoherence, yielding then the
optimal quantum-enhancement. We are able to do so by constructing a
noise-induced lower bound on the error that stems from a general method of
classically simulating a noisy quantum evolution, during which the stochastic
parameter to be estimated -- here, the magnetic field -- is encoded. The method
naturally extends to schemes beyond the linear-Gaussian regime, in particular,
also to ones involving feedback or active control.
