Distribution of quantum Fisher information in asymmetric cloning machines

An unknown quantum state cannot be copied on demand and broadcast freely due to the famous no-cloning theorem. Approximate cloning schemes have been proposed to achieve the optimal cloning characterized by the maximal fidelity between the original and its copies. Here, from the perspective of quantum Fisher information (QFI), we investigate the distribution of QFI in asymmetric cloning machines which produce two nonidentical copies. As one might expect, improving the QFI of one copy results in decreasing the QFI of the other copy, roughly the same as that of fidelity. It is perhaps also unsurprising that asymmetric phase-covariant cloning machine outperforms universal cloning machine in distributing QFI since a priori information of the input state has been utilized. However, interesting results appear when we compare the distributabilities of fidelity (which quantifies the full information of quantum states), and QFI (which only captures the information of relevant parameters) in asymmetric cloning machines. In contrast to the results of fidelity, where the distributability of symmetric cloning is always optimal for any d-dimensional cloning, we find that asymmetric cloning performs always better than symmetric cloning on the distribution of QFI for $d\leq18$, but this conclusion becomes invalid when $d>18$.