Action Quantum Speed Limits

Quantum speed limits (QSLs) place a lower bound on the evolution time of a quantum system between two states. These bounds are well understood for pure states evolving under unitary dynamics. In the case of non-unitary dynamics and mixed states, there is no general consensus on which bound is the best or even how to properly interpret them. We introduce action 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. Employing techniques from optimal control theory we demonstrate the attainablitity of these bounds for the case of a thermalizing qubit. 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.