  We introduce action quantum speed limits (QSLs) as a family of bounds on the
minimal time to connect two states that, unlike the usual geometric approach,
crucially depend on how the path is traversed, i.e. on the instantaneous speed.
The two approaches provide consistent bounds when the instantaneous speed is
optimized along a fixed path and we demonstrate this explicitly for the case of
a thermalizing qubit employing techniques from optimal control theory. In
addition, we critically analyze the interpretation of QSLs based on different
choices of metric establishing that, in general, these open system QSL times
provide an indication of optimality with respect to the geodesic path, rather
than necessarily being indicative of an achievable minimal time.
