Quantum evolution speed induced by Hamiltonian

Abstract

In the issue of quantum evolution, quantum evolution speed is usually quantified by the time rate of change of state distance between the initial sate and its time evolution. In this paper, the path distance of quantum evolution is introduced to study the evolution of a quantum system, through the approach combined with basic theory of quantum evolution and the linear algebra. In a quantum unitary system, the quantum evolution operator contains the path information of the quantum evolution, where the path distance is determined by the principal argument of the eigenvalues of the unitary operator. Accordingly, the instantaneous quantum evolution speed is proportional to the distance between the maximum and minimum eigenvalues of the Hamiltonian. As one of the applications, the path distance and the instantaneous quantum evolution speed could be used to form a new lower bound of the real evolution time, which depends on the evolution operator and Hamiltonian, and is independent of the initial state. It is found that the lower bound presented here is exactly equal to the real evolution time in the range <inline-formula><tex-math id="M2">\begin{document}$ \left[ {0, {\pi }/({{2{\omega _{\rm{H}}}}}}) \right]$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231009_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231009_M2.png"/></alternatives></inline-formula>. The tool of path distance and instantaneous quantum evolution speed introduced here provides new method for the related researches.