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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), theinformation-extraction bottleneck can stem from high post-processing costs ordetector saturation. In these regimes, it is desirable to compress theinformation 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 fullycharacterize 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 weshow is optimal for qubit probes. Further, we show that, for the optimal filterin this family, compression is still lossless if noise acts after the filter.However, in the presence of depolarizing noise before filtering, compression islossy. In both cases, information-extraction can be implemented significantlybetter than simply discarding a constant fraction of the states, even in thepresence of strong noise.

Compression of metrological quantum information in the presence of noise