Quantitative conditions for time evolution in terms of the von Neumann equation

• 来源期刊: 中国科学:物理学 力学 天文学（英文版）
• 年，卷（期）: 2018,（7）
• 页码范围: 54-60
• 页数: 7页
• 语种: 英文

摘要

The adiabatic theorem describes the time evolution of the pure state and gives an adiabatic approximate solution to the Schro¨dinger equation by choosing a single eigenstate of the Hamiltonian as the initial state. In quantum systems, states are divided into pure states (unite vectors) and mixed states (density matrices, i.e., positive operators with trace one). Accordingly, mixed states have their own corresponding time evolution, which is described by the von Neumann equation. In this paper, we discuss the quantitative conditions for the time evolution of mixed states in terms of the von Neumann equation. First, we introduce the definitions for uniformly slowly evolving and δ-uniformly slowly evolving with respect to mixed states, then we present a necessary and sufficient condition for the Hamiltonian of the system to be uniformly slowly evolving and we obtain some upper bounds for the adiabatic approximate error. Lastly, we illustrate our results in an example.