Family of attainable geometric quantum speed limits

We propose a quantum state distance and develop a family of geometrical quantum speed limits (QSLs) for open and closed systems. The QSL time includes an alternative function by which we derive three QSL times with particularly chosen functions. It indicates that two QSL times are exactly the ones presented in Ref. \cite{Campaioli2019tightrobust} and \cite{mai2023tight}, respectively, and the third one can provide a unified QSL time for both open and closed systems. The three QSL times are attainable for any given initial state in the sense that there exists a dynamics driving the initial state to evolve along the geodesic. We numerically compare the tightness of the three QSL times, which typically promises a tighter QSL time if optimizing the alternative function.