Maximal quantum Fisher information matrix

Westudy the existence of the maximal quantum Fisher information matrix in the multi-parameter quantum estimation, which bounds the ultimate precision limit. Weshow that when the maximal quantum Fisher information matrix exists, it can be directly obtained from the underlying dynamics. Examples are then provided to demonstrate the usefulness of the maximal quantum Fisher information matrix by deriving various trade-off relations in multi-parameter quantum estimation and obtaining the bounds for the scalings of the precision limit.