We acknowledge that a derivation reported in Phys. Rev. Lett. 125, 040601 (2020) is incorrect as pointed out by Cusumano and Rudnicki. We respond by giving a correct proof of the claim “fluctuations in the free energy operator upper bound the charging power of a quantum battery” that we made in the Letter.

1 aGarcía-Pintos, Luis, Pedro1 aHamma, Alioscia1 adel Campo, Adolfo uhttps://quics.umd.edu/publications/garc%C3%ADa-pintos-hamma-and-del-campo-reply01577nas a2200157 4500008004100000245006300041210006300104260001300167520107800180100003201258700002401290700002101314700002201335700002501357856003701382 2021 eng d00aUnifying Quantum and Classical Speed Limits on Observables0 aUnifying Quantum and Classical Speed Limits on Observables c8/9/20213 aThe 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 divides into Mandalestam and Tamm's original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, generalizing both to open quantum systems. By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds to 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 \emph{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.

1 aGarcía-Pintos, Luis, Pedro1 aNicholson, Schuyler1 aGreen, Jason, R.1 adel Campo, Adolfo1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/2108.0426101237nas a2200145 4500008004100000245008300041210006900124260001400193490000800207520076500215100003200980700002001012700002201032856003701054 2020 eng d00aFluctuations in Extractable Work Bound the Charging Power of Quantum Batteries0 aFluctuations in Extractable Work Bound the Charging Power of Qua c7/22/20200 v1253 aWe study the connection between the charging power of quantum batteries and the fluctuations of the stored work. We prove that in order to have a non-zero rate of change of the extractable work, the state ρW of the battery cannot be an eigenstate of a `\emph{work operator}', defined by F ≡ HW + β−1log(ρW), where HW is the Hamiltonian of the battery and β is the inverse temperature of a reference thermal bath with respect to which the extractable work is calculated. We do so by proving that fluctuations in the stored work upper bound the charging power of a quantum battery. Our findings also suggest that quantum coherence in the battery enhances the charging process, which we illustrate on a toy model of a heat engine.

1 aGarcía-Pintos, Luis, Pedro1 aHamma, Alioscia1 adel Campo, Adolfo uhttps://arxiv.org/abs/1909.0355801726nas a2200145 4500008004100000245006100041210006000102260001500162520126300177100002801440700003201468700002201500700002101522856003701543 2020 eng d00aTime-information uncertainty relations in thermodynamics0 aTimeinformation uncertainty relations in thermodynamics c09/21/20203 aPhysical systems that power motion and create structure in a fixed amount of time dissipate energy and produce entropy. Whether living or synthetic, 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 initial standard deviation. From this general bound, we establish a family of speed limits for heat, work, entropy production, and entropy flow depending on the experimental constraints on the system. In all of these inequalities, the time scale 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 assumptions about the stochastic dynamics or their function.

1 aNicholson, Schuyler, B.1 aGarcía-Pintos, Luis, Pedro1 adel Campo, Adolfo1 aGreen, Jason, R. uhttps://arxiv.org/abs/2001.05418