A Brief History of Quantum Speed Limits in Isolated Systems

Rikuya Kobashi

Niklas Hörnedal, Ole Sönnerborn

Many quantum speed limits for isolated systems can be generalized to also apply to closed systems. This is, for example, the case with the well-known Mandelstam-Tamm quantum speed limit. Margolus and Levitin derived an equally well-known and ostensibly related quantum speed limit, and it seems to be widely believed that the Margolus-Levitin quantum speed limit can be similarly generalized to closed systems. However, a recent geometrical examination of this limit reveals that it differs significantly from most known quantum speed limits. In this paper, we show that, contrary to the common belief, the Margolus-Levitin quantum speed limit does not extend to closed systems in an obvious way. More precisely, we show that for every hypothetical bound of Margolus-Levitin type, there are closed systems that evolve with a conserved normalized expected energy between states with any given fidelity in a time shorter than the bound. We also show that for isolated systems, the Mandelstam-Tamm quantum speed limit and a slightly weakened version of this limit that we call the Bhatia-Davies quantum speed limit always saturate simultaneously. Both of these evolution time estimates extend straightforwardly to closed systems. We demonstrate that there are closed systems that saturate the Mandelstam-Tamm but not the Bhatia-Davies quantum speed limit.

Michael R. Frey

Luis Pedro García-Pintos, Schuyler B. Nicholson, Jason R. Green +2 more

The presence of noise or the interaction with an environment can radically change the dynamics of observables of an otherwise isolated quantum system.We derive a bound on the speed with which observables of open quantum systems evolve.This speed limit is divided into Mandelstam and Tamm's original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, and both are generalized to open quantum systems.By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds on the speed of evolution.We prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits and that a preferred basis of speed operators serves to completely characterize the observables that saturate the speed limits.We use this construction to bound the effect of incoherent dynamics on the evolution of an observable and to find the Hamiltonian that gives the maximum coherent speedup to the evolution of an observable.