Research On Theory Of Quantum Adiabatic Evolution And Dynamical Properties Of Separable States

As a core tool of quantum theory, quantum adiabatic theorem reflects the evolution of the quantum system with time-dependent Hamiltonian and provides an approximate solution to the Schrodinger equation. Recently, the inconsistency in the ap-plication of the adiabatic theorem has caused concern based on the conventional adiabatic condition. The entanglement or separability of quantum state reflects the intrinsic prop-erty of the state and the entanglement state is considered to be an important resource in quantum information processing. This dissertation is devoted to research on the theory of quantum adiabatic evolution and dynamical properties of separable states by the methods of operator theory and matrix analysis theory. The aim of this dissertation is to establish the precise traditional quantum adiabatic theorem, estimate the adiabatic approximation error, propose the generalized adiabatic evolution concepts, as well as the non-Hermitian adiabatic evolution and adiabatic approximation theorem, and discuss the dynamical prop-erties of separable states by finding some Hamiltonians that preserve the separability of the initial state.In Chapter 1, we introduce some research backgrounds and status on our main con-tents, and list some relevant notations, definitions and theorems.In Chapter 2, we study the quantum adiabatic evolution and adiabatic approximation theory. Firstly, we introduce some notations about the quantum adiabatic evolution, and obtain a necessary and sufficient condition for the conventional quantum adiabatic evolu-tion. Based on the spectral decomposition of Hermitian operators, we choose the ground states, which are orthogonal with their derivatives, obtain an upper bound for the adia-batic approximation error where the error is estimated by the difference and the fidelity between the exact solution and the adiabatic approximation solution to the Schrodinger equation, respectively. In particular, under the gap condition, we get another upper bound for the adiabatic approximation error, and obtain a sufficient condition for the adiabatic approximation. Finally, our results are illustrated by an example.In Chapter 3, we discuss the generalized quantum adiabatic evolution and adiabatic approximation theory. Based on the superposition principle of quantum states, we de-fine the generalized adiabatic approximate solution, introduce the generalized quantum adiabatic evolution and the generalized adiabatic approximation, and get a necessary and sufficient condition for the generalized quantum adiabatic evolution. We give an upper bound of generalized adiabatic approximation error. Finally, we verify the correctness of the conclusion by comparing the exact solution and approximate solution with an exam-ple.In Chapter 4, we investigate the quantum adiabatic evolution and adiabatic approx-imation theory in the non-Hermitian quantum system. We introduce the definitions of generalized fidelity, A-uniformly slowly evolving, and δ-A-uniformly slowly evolving and obtain a necessary and sufficient condition for the non-Hermitian adiabatic evolu-tion. Then we discuss the relationship between the solutions to the Schrodinger equation determined by the Hermitian Hamiltonian and non-Hermitian Hamiltonian, respectively, and explore the upper bound for non-Hermitian adiabatic approximation error. In ad-dition, we obtain the same upper bound by constructing a new inner product, and get a sufficient condition for the δ-A-uniformly slowly evolving. Finally, we use an example to illustrate the theorem.In Chapter 5, we discuss the dynamical properties of separable states. In bipartite quantum system, according to the methods of functional calculus and Taylor expansions, we obtain four kinds of Hamiltonians that preserve the separability of initial states.