Several tasks in quantum-information processing involve quantum learning. For example, quantum sensing, quantum machine learning and quantum-computer calibration involve learning and estimating unknown parameters $\bar{\theta} = (\theta_1, \theta_2,..., \theta_M)$ from measurements of many copies of a quantum state $\rho_{\bar{\theta}}$. This type of metrological information is described by the quantum Fisher information matrix, which bounds the average amount of information learnt about $\bar{\theta}$ per measurement of $\rho_{\bar{\theta}}$. In this talk, I will show that the quantum Fisher information about multiple parameters encoded in ${\rho_{\bar{\theta}}}^{\otimes N}$ can be compressed into ${\rho^{\star}_{\bar{\theta}}}^{\otimes M}$, were $M < N$. I will show that $M/N$ can be made arbitrarily small, and that the compression can happen without loss of information. Finally, I will demonstrate how to construct filters that perform this unbounded and lossless information compression. Our results are not only theoretically interesting, but also practically. In several technologies, it is advantageous to compress information in as few states as possible, for example, to avoid detector saturation and/or to reduce post-processing costs. Our filters can reduce arbitrarily the quantum-state intensity on experimental detectors, while retaining all initial information.