We develop an intuitive geometric picture of quantum states, define a
particular state distance, and derive a quantum speed limit (QSL) for open
systems. Our QSL is attainable because any initial state can be driven to a
final state by the particular dynamics along the geodesic. We present the
general condition for dynamics along the geodesic for our QSL. As evidence, we
consider the generalized amplitude damping dynamics and the dephasing dynamics
to demonstrate the attainability. In addition, we also compare our QSL with
others by strict analytic processes as well as numerical illustrations, and
show our QSL is tight in many cases. It indicates that our work is significant
in tightening the bound of evolution time