Compression of metrological quantum information in the presence of noise

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.