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. [1] and [2], 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