Compression of metrological quantum information in the presence of noise

In quantum metrology, information about unknown parameters $\mathbfθ = (θ_1,\ldots,θ_M)$ is accessed by measuring probe states $\hatρ_{\mathbfθ}$. In experimental settings where copies of $\hatρ_{\mathbfθ}$ 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 $\hatρ_{\mathbfθ} \, ^{\otimes n}$ into $m