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{\textquoteright}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.

}, keywords = {limits, Observables}, doi = {10.1103/PhysRevX.12.011038}, url = {https://link.aps.org/doi/10.1103/PhysRevX.12.011038}, author = {Garcia-Pintos, Luis Pedro and Nicholson, Schuyler B. and Green, Jason R. and del Campo, Adolfo and Gorshkov, Alexey V.} } @article {nicholson_time-information_2020, title = {Time-information uncertainty relations in thermodynamics}, journal = {Nat. Phys.}, volume = {16}, number = {12}, year = {2020}, note = {Place: HEIDELBERGER PLATZ 3, BERLIN, 14197, GERMANY Publisher: NATURE RESEARCH Type: Article}, month = {dec}, abstract = {Physical systems powering motion and creating structure in a fixed amount of time dissipate energy and produce entropy. Whether living, synthetic or engineered, systems performing these dynamic functions must balance dissipation and speed. Here, we show that rates of energy and entropy exchange are subject to a speed limit-a time-information uncertainty relation-imposed by the rates of change in the information content of the system. This uncertainty relation bounds the time that elapses before the change in a thermodynamic quantity has the same magnitude as its s.d. From this general bound, we establish a family of speed limits for heat, dissipated/chemical work and entropy depending on the experimental constraints on the system and its environment. In all of these inequalities, the timescale of transient dynamical fluctuations is universally bounded by the Fisher information. Moreover, they all have a mathematical form that mirrors the Mandelstam-Tamm version of the time-energy uncertainty relation in quantum mechanics. These bounds on the speed of arbitrary observables apply to transient systems away from thermodynamic equilibrium, independent of the physical constraints on the stochastic dynamics or their function. A time-information uncertainty relation in thermodynamics has been derived, analogous to the time-energy uncertainty relation in quantum mechanics, imposing limits on the speed of energy and entropy exchange between a system and external reservoirs.}, issn = {1745-2473}, doi = {10.1038/s41567-020-0981-y}, author = {Nicholson, Schuyler B. and Garcia-Pintos, Luis Pedro and del Campo, Adolfo and Green, Jason R.} }