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arXiv:1302.5069 · Physical Review Letters

Quantum Speed Limit for Non-Markovian Dynamics

Sebastian Deffner, Eric Lutz

Published Jul 3, 2013• 510 citations
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Abstract

We derive a Margolus-Levitin-type bound on the minimal evolution time of an arbitrarily driven open quantum system. We express this quantum speed limit time in terms of the operator norm of the nonunitary generator of the dynamics. We apply these results to the damped Jaynes-Cummings model and demonstrate that the corresponding bound is tight. We further show that non-Markovian effects can speed up quantum evolution and therefore lead to a smaller quantum speed limit time.

DOI
10.1103/physrevlett.111.010402

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We derive a Margolus-Levitin-type bound on the minimal evolution time of an arbitrarily driven open quantum system. We express this quantum speed limit time in terms of the operator norm of the nonunitary generator of the dynamics. We apply these results to the damped Jaynes-Cummings model and demonstrate that the corresponding bound is tight. We further show that non-Markovian effects can speed up quantum evolution and therefore lead to a smaller quantum speed limit time.
Quantum mechanics dictates bounds for the minimal evolution time between predetermined initial and final states. Several of these quantum-speed-limit (QSL) bounds were derived for nonunitary dynamics using different approaches. Here, we perform a systematic analysis of the most common QSL bounds in the damped Jaynes-Cummings model, covering the Markovian and non-Markovian regimes. We show that only one of the analyzed bounds cleaves to the essence of the QSL theory outlined in the pioneer works of Mandelstam and Tamm and of Margolus and Levitin in the context of unitary evolutions. We also show that all QSL bounds analyzed reflect the fact that in our model non-Markovian effects speed up quantum evolution. However, it is not possible to infer Markovian or non-Markovian behavior of the dynamics by analyzing only the QSL bounds.
The minimal time a system needs to evolve from an initial state to its one orthogonal state is defined as the quantum speed limit time, which can be used to characterize the maximal speed of evolution of a quantum system. This is a fundamental question of quantum physics. We investigate the generic bound on the minimal evolution time of the open dynamical quantum system. This quantum speed limit time is applicable to both mixed and pure initial states. We then apply this result to the damped Jaynes-Cummings model and the Ohimc-like dephasing model starting from a general time-evolution state. The bound of this time-dependent state at any point in time can be found. For the damped Jaynes-Cummings model, when the system starts from the excited state, the corresponding bound first decreases and then increases in the Markovian dynamics. While in the non-Markovian regime, the speed limit time shows an interesting periodic oscillatory behavior. For the case of Ohimc-like dephasing model, this bound would be gradually trapped to a fixed value. In addition, the roles of the relativistic effects on the speed limit time for the observer in non-inertial frames are discussed.
We theoretically study the dynamical speedup of a quantum system in a nonequilibrium environment. Based on the trace distance, we derive the generalized Margolus-Levitin and Mandelstam-Tamm types of bounds on the quantum speed limit time of a quantum system evolving from an arbitrary initial state. We demonstrate that the mechanism for the speedup of dynamical evolution is closely associated with both the energy of the system and exchange of information between the system and its environment. It is shown that the nonequilibrium feature of the environment can speed up the quantum evolution in both Markovian and non-Markovian dynamics regions. We emphasize that the non-Markovian effect of the system dynamics is neither necessary nor sufficient to speed up the quantum evolution in a nonequilibrium environment.
Abstract Quantum theory sets a fundamental bound for the minimal time evolution between initial and final state of a system . This bound is called quantum speed limit time , which is used to quantify the speed of evolution of a quantum system. If quantum speed limit time decreases, the evolution keeps on increasing. In this paper, we study the effect of quantum speed limit time in terms of correlated and uncorrelated noise channels like amplitude damping and phase damping channel. We use maximally entangled mixed state as our initial state. We observe the behavior of quantum speed limit time under Markovian and non-Markovian dynamics. We prove that for the special class of our inital state and for the dynamical evolution of a system, there exist a link between quantum speed limit time and non-Markovianity. However, an interesting results were also observed regarding the tightness of bound in Markovian region. Our results shed light on the information back-flow between system and environment in terms of quantum evolution which is useful as a resource in quantum information processing task.
Quantum speed limit (QSL) time under noise has drawn considerable attention in real quantum computational processes. Though non-Markovian noise is found to be able to accelerate quantum evolution for a damped Jaynes—Cummings model, in this work we show that non-Markovianity will slow down the quantum evolution of an experimentally controllable photon system. As an application, QSL time of a photon can be controlled by regulating the relevant environment parameter properly, which nearly reaches the currently available photonic experimental technology.
A unified bound on the quantum speed limit is obtained for open quantum systems with a mixed initial state by utilizing the function of relative purity proposed by F. Campaioli et al. [Phys. Rev. Lett. 120, 060409 (2018)]. As applications, it is found that the quantum speed limit bound for the damped Jaynes-Cummings model is determined by the competition among the non-Markovianity, the population of the initial excited state, and the initial-state coherence, which shows that the population of the initial excited state and the coherence of the initial state can make the quantum speed limit bound tighter even though the non-Markovian effects can accelerate the evolution of open quantum system. For the dephasing model, a simple factorization law with the initial-state coherence shows that the quantum speed limit is governed by only the competition between the non-Markovianity and the coherence of the initial state.