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arXiv:1607.00941 · Europhysics Letters (EPL)

Speed limits in Liouville space for open quantum systems

Raam Uzdin, Ronnie Kosloff

Published Aug 1, 2016• 52 citations
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Abstract

One of the defining properties of an open quantum system is the variation of its purity in time. We derive speed limits on the rate of purity change for systems coupled to a Markovian environment. Our speed limits are based on Liouville space where density matrices are represented as vectors. This approach leads to speed limits that are always tighter compared to their parallel speed limits in Hilbert space. These bounds depend solely on the generators of the nonunitary dynamics and are independent of the particular state of the systems. Thus, they are perfectly suited to investigate dephasing, thermalization, and decorrelation processes of arbitrary states. We show that our speed limits can be attained and are therefore tight. As an application of our results we study dephasing of interacting spins, and the speed of classical and quantum correlation erasure in multi-particle system.

DOI
10.1209/0295-5075/115/40003

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One of the defining properties of an open quantum system is the variation of its purity in time. We derive speed limits on the rate of purity change for systems coupled to a Markovian environment. Our speed limits are based on Liouville space where density matrices are represented as vectors. This approach leads to speed limits that are always tighter compared to their parallel speed limits in Hilbert space. These bounds depend solely on the generators of the nonunitary dynamics and are independent of the particular state of the systems. Thus, they are perfectly suited to investigate dephasing, thermalization, and decorrelation processes of arbitrary states. We show that our speed limits can be attained and are therefore tight. As an application of our results we study dephasing of interacting spins, and the speed of classical and quantum correlation erasure in multi-particle system.
Starting from a geometric perspective, we derive a quantum speed limit for arbitrary open quantum evolution, which could be Markovian or non-Markovian, providing a fundamental bound on the time taken for the most general quantum dynamics. Our methods rely on measuring angles and distances between (mixed) states represented as generalized Bloch vectors. We study the properties of our bound and present its form for closed and open evolution, with the latter in both Lindblad form and in terms of a memory kernel. Our speed limit is provably robust under composition and mixing, features that largely improve the effectiveness of quantum speed limits for open evolution of mixed states. We also demonstrate that our bound is easier to compute and measure than other quantum speed limits for open evolution, and that it is tighter than the previous bounds for almost all open processes. Finally, we discuss the usefulness of quantum speed limits and their impact in current research.
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.
We theoretically study the quantum speed limits (QSLs) of a qubit system coupled to a thermal dephasing environment with an Ohmic-like spectral density. Based on the geometric QSLs time bound, which is derived by employing the trace distance to quantify the geodesic between two distinguishable states in dynamical evolution, we study the influences of the temperature and spectral density of the environment on the QSLs time of the dephasing qubit. We also investigate the interplay between the QSLs time, the environmental temperature, and the spectral density of the environment. It has been demonstrated that the QSLs time closely depends on the transition frequency and the dynamical behavior (e.g., coherence trapping) of the dephasing qubit. For a fixed Ohmicity parameter of the environmental spectral density, the increase in environmental temperature can enhance the QSLs time bound. In addition, when the environmental temperature remains constant, the increase in the Ohmicity parameter initially leads to a reduction in the QSLs time bound, which is then followed by an increase in the time bound of QSLs. Our results can help to better understand the QSLs in the dynamics of open quantum systems and have potential application in the modulation of QSLs time in the dephasing qubit by engineering the spectral density of the environment.
We introduce the concept of magic subspaces for the control of dissipative NNN- level quantum systems whose dynamics are governed by the Lindblad equation. For a given purity, these subspaces can be defined as the set of density matrices for which the rate of purity change is maximum or minimum. Adding fictitious control fields to the system so two density operators with the same purity can be connected in a very short time, we show that magic subspaces allow us to derive a purity speed limit, which only depends on the relaxation rates. We emphasize the superiority of this limit with respect to established bounds and its tightness in the case of a two-level dissipative quantum system. The link between the speed limit and the corresponding time-optimal solution is discussed in the framework of this study. Explicit examples are described for two- and three-level quantum systems.
Traditional quantum speed limits formulated in density matrix space are generally unattainable for a wide class of dynamics and it is difficult to characterize the fastest possible dynamics. To address this, we present two distinct quantum speed limits in Liouville space for completely positive and trace-preserving (CPTP) dynamics. The first bound saturates for time-optimal CPTP dynamics, while the second bound is exact for all states and all CPTP dynamics. Our bounds have a clear physical and geometric interpretation arising from the uncertainty relations for operators acting on Liouville space, and the geometry of quantum evolution in Liouville space. We also obtain the form of the Liouvillian, which generates the time-optimal CPTP dynamics that connects the given initial and target states. To illustrate our findings, we show that the speed of evolution in Liouville space bounds the growth of the spectral form factor and Krylov complexity of states, which are crucial for studying information scrambling and quantum chaos. In another important application, we show that our results can help us understand the counterintuitive phenomenon of the Mpemba effect in nonequilibrium open quantum dynamics, as the minimal relaxation time scale obtained by speed limits is dictated by the eigenmodes of the Liouvillian.