Quantum limits to dynamical evolution

How fast can a quantum system evolve in time, given a certain amount of energy? If the system is composed of a number of subsystems, is entanglement a useful resource in speeding up the dynamical evolution? To answer the first of these two questions, one typically defines some characteristic time of the dynamics and studies its connections with the energy resources of the initial state of the system. Most of the previous results in this field @1–3# trace back to the timeenergy uncertainty relation in the form derived by Mandelstam and Tamm @4#: in this way, the various lifetimes are bounded by the energy spread DE of the system. More recently, however, Margolus and Levitin pointed out that one can relate the characteristic times of the system also to the average energy E of the initial state @5#. In particular, defining the lifetime of the system as the time it takes for it to evolve to an orthogonal state, the above results allow one to introduce a quantum speed limit time as the minimum possible lifetime for a system of average energy E and spread DE . In @6# we analyzed such a bound in the case of composite systems ~i.e., systems composed of a collection of subsystems!. In this paper we extend these results by analyzing what happens when the quantum speed limit time is generalized by redefining it as the minimum time t it takes for the initial state % to evolve through a unitary evolution to a state %(t) such that the fidelity F„% ,%(t)... of @7# is equal to a given e P@0,1# . Even though the scenario is more complex than the case e50 of @6#, in this case also it is possible to show that entanglement is useful to achieve speedup of the dynamics if one wants to share the energy resources among the subsystems. In Sec. I we extend the definition of quantum speed limit time and derive its expression in terms of the energy characteristics of the initial state, first considering the case of pure states and then extending the analysis to the more complex case of nonpure states. In Sec. II we analyze the role that entanglement among subsystems plays in achieving the quantum speed limit. Most of the technical details of the derivations have been inserted in the Appendix.