Attainable quantum speed limit for N-dimensional quantum systems

Abstract

Quantum speed limit (QSL) is a fundamental concept in quantum mechanics and provides a lower bound on the evolution time. The attainability of QSL, greatly depending on the understanding of QSL, is a long-standing open problem especially for high-dimensional systems. In this paper, we solve this problem by establishing a QSL suitable and attainable for both open and closed quantum systems based on a new proposed state distance. It is shown that given any initial state in a certain dimension, our QSL bound can always be saturated by unitary and non-unitary dynamics, and for any given Hamiltonian for a unitary evolution, a pair of states always exists, saturating the bound. As applications, we demonstrate the QSL time attained by various physical settings. This paper will shed new light on the QSL problems.

Figures (2)

• Figure 1: (Left)The QSL time $\tau_{\mathrm{QSL}}$ versus the actual unitary evolution time $\tau$. The red line corresponds to the optimal Hamiltonian $H_T$, and the blue line means $H=H_0+H_1$. Here $E_m=1$, $\Omega= E_m$ and $\mu=0.5$. (Right)QSL versus evolution time (blue line). The time-dependent decay rate (red line) is shown as the right axis. $\omega_c=1$ and $k=4$ are taken for the decay rate in the non-Markovian regime.
• Figure 2: Different initial state vs corresponding QSL. The evolution time is $\tau=1$. The system is prepared as excited state initially with $\lambda_0=0$ (Left), and the ground state is partly occupied with $\lambda_0=0.5$ (Right), respectively.