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arXiv:1906.04766 · Physical Review Research

Evolution speed of open quantum dynamics

Dorje C. Brody, Bradley Longstaff

Published Nov 25, 2019• 32 citations
arXivHTMLPDF

Abstract

The authors present a theory to address the evolution speed for open-system dynamics and show that time-optimal control of quantum state hinges on understanding the interplay between the environmental influences, that may be beyond control, and the internal ``unitary'' dynamics.
DOI
10.1103/physrevresearch.1.033127

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The average energy and energy spread of a quantum system are two important factors that affect a system's evolution speed, which is usually defined as the changing rate of the distance between a quantum state and the state during its evolution. Alternatively, the evolution speed of a quantum system could also be studied based on its evolution path. It is found that if we define the evolution speed of a quantum unitary system as the changing rate of the path distance between a quantum state and the state during its evolution, the instantaneous evolution speed is determined by the Hamiltonian's eigenvalues, rather than its expectation value or variance imposed on any quantum state. More specifically, the range of the Hamiltonian's eigenvalues determines the evolution speed of a quantum system through the relation ${\ensuremath{\omega}}_{H}=\frac{1}{2}({\ensuremath{\lambda}}_{max}\ensuremath{-}{\ensuremath{\lambda}}_{min})$, which is independent of the quantum state. Given an initial state ${\ensuremath{\rho}}_{0}$ in a quantum unitary system, the minimum time required to evolve it to a target state ${\ensuremath{\rho}}_{\ensuremath{\tau}}$ is ${\ensuremath{\tau}}_{min}=\frac{{\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})}{{\ensuremath{\omega}}_{H}}$, where ${\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})$ is the path distance contained in the shortest evolution from ${\ensuremath{\rho}}_{0}$ to ${\ensuremath{\rho}}_{\ensuremath{\tau}}$. This lower bound of the evolution time is tight and achievable in a finite-energy system.
The space of density matrices is embedded in a Euclidean space to deduce the dynamical equation \nsatisfied by the state of an open quantum system. The Euclidean norm is used to obtain an explicit \nexpression for the speed of the evolution of the state. The unitary contribution to the evolution \nspeed is given by the modified skew information of the Hamiltonian, while the radial component of \nthe evolution speed, connected to the rate at which the purity of the state changes, is shown to be \ndetermined by the modified skew information of the Lindblad operators. An open-system analogue \nof the quantum navigation problem is posed, and a perturbative analysis is presented to identify \nthe amount of change on the speed. Properties of the evolution speed are examined further through \nexample systems, showing that the evolution speed need not be a decreasing function of time.
In this chapter we shall deal with the quantum dynamics of an open system. By open quantum system we mean here a system which interacts with an environment (see Sec. 9.4): since we are not interested in the dynamics of the environment, we shall have to describe the evolution of the system in some “effective” way. In particular, if we consider only the evolution of the system, it will be non-unitary, and this will represent the subject of this chapter. As we know, Hamiltonian quantum dynamics is unitary and changes pure states into pure states. On the other hand, non-unitary dynamics changes an initially pure state into a mixture, which must be described by a density matrix (see Ch. 5). In the case of macroscopic systems, the coupling with the environment may be arbitrarily reduced and therefore its influence can be made correspondingly small (see Sec. 1.1). Microscopic systems, however, always couple to the environment and this coupling cannot be considered negligible. This is the reason why the quantum theory of open systems is one of the most important and fundamental chapters of quantum mechanics that, though born in quantum optics, has many implications in almost all fields of physics. The present chapter can be seen as a further development of the measurement theory (see Ch. 9), as open systems manifest a decoherent dynamics (see in particular Sec. 9.4).
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.
The notion of a quantum speed limit (QSL) refers to the fundamental fact that two quantum states become completely distinguishable upon dynamical evolution only after a finite amount of time, called the QSL time. A different but related concept is that of minimum control time (MCT), which is the minimum evolution time needed for a state to be driven (by suitable, generally time-dependent, control fields) to a given target state. While the QSL can give information about the MCT, it usually imposes little restrictions on it and is thus impractical for control purposes. In this work, we revisit this issue by first presenting a theory of geometrical QSL for unitary transformations, rather than for states, and discuss its implications and limitations. Then, we propose a framework for bounding the MCT for realizing unitary transformations that goes beyond the QSL results and gives much more meaningful information to understand the controlled dynamics of the system at short times.