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在研究量子态的演化问题时, 量子演化速率往往定义为量子初态与其演化态之间的态距离随时间的变化率. 本文将量子演化的基本理论与线性代数方法相结合, 通过量子态演化的路径距离来研究量子系统的演化. 量子幺正演化系统中, 量子演化算符包含了量子演化的路径信息, 路径距离的大小则决定于演化算符本征值的幅角主值分布. 由量子态演化的路径距离随时间的变化率而得到的量子瞬时演化速率则正比于系统哈密顿量的最大与最小本征值之差. 作为应用之一, 利用量子演化的路径距离及哈密顿量诱导的瞬时演化速率, 可以给出量子演化新的时间下限. 此时间下限只与系统的演化算符及哈密顿量有关, 而与量子初态的具体形式无关, 这与量子系统真实演化时间所具有的性质一致. 严格的理论证明以及两个演化实例的数值结果均表明, 在
$ [0,\ \pi/(2\omega_{\mathrm{H}})]$ 时间范围内, 本文给出的演化时间下限与真实演化时间重合, 是真实演化时间的准确预测. 通过量子演化的路径距离及相应演化速率来研究量子系统的演化, 为相关问题的解答提供了新的思路和方法.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$ \left[ {0, {\pi }/({{2{\omega _{\rm{H}}}}}}) \right]$ . The tool of path distance and instantaneous quantum evolution speed introduced here provides new method for the related researches.-
Keywords:
- quantum evolution/
- path distance/
- quantum evolution speed
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