  In quantum metrology, information about unknown parameters $\mathbf{\theta} =
(\theta_1,\ldots,\theta_M)$ is accessed by measuring probe states
$\hat{\rho}_{\mathbf{\theta}}$. In experimental settings where copies of
$\hat{\rho}_{\mathbf{\theta}}$ can be produced rapidly (e.g., in optics), the
information-extraction bottleneck can stem from high post-processing costs or
detector saturation. In these regimes, it is desirable to compress the
information encoded in $\hat{\rho}_{\mathbf{\theta}} \, ^{\otimes n}$ into
$m<n$ copies of a postselected state:
${\hat{\rho}_{\mathbf{\theta}}^{\text{ps}}} \,^{\otimes m}$.
  Remarkably, recent works have shown that, in the absence of noise,
compression can be lossless, for $m/n$ arbitrarily small. Here, we fully
characterize the family of filters that enable lossless compression. Further,
we study the effect of noise on quantum-metrological information amplification.
Motivated by experiments, we consider a popular family of filters, which we
show is optimal for qubit probes. Further, we show that, for the optimal filter
in this family, compression is still lossless if noise acts after the filter.
However, in the presence of depolarizing noise before filtering, compression is
lossy. In both cases, information-extraction can be implemented significantly
better than simply discarding a constant fraction of the states, even in the
presence of strong noise.

Quantum Physics