Numerical Ranges and Applications in Quantum Information

Abstract

The numerical range (NR) of a matrix is a concept that first arose in the early 1900’s as part of efforts to build a rigorous mathematical framework for quantum mechanics and other challenges at the time. Quantum information science (QIS) is a new field having risen to prominence over the past two decades, combining quantum physics and information science. In this thesis, we connect NR and its extensions with QIS in several topics and apply the knowledge to solve related QIS problems. We utilize the insight offered by NRs and apply them to quantum state inference and Hamiltonian reconstruction. We propose a new general deep learning method for quantum state inference in chapter 3 and employ it to two scenarios – maximum entropy estimation of unknown states and ground state reconstruction. The method manifests high fidelity, exceptional efficiency, and noise tolerance on both numerical and experimental data. A new optimization algorithm is presented in chapter 4 for recovering Hamiltonians. It takes in partial measurements from any eigenstates of an unknown Hamiltonian H, then provides an estimation of H. Our algorithm almost perfectly deduces generic and local generic Hamiltonians. Inspired by hybrid classical-quantum error correction (Hybrid QEC), the higher rank (k : p)-matricial range is defined and studied in chapter 5. This concept is a new extension of NR. We use it to study Hybrid QEC, also to investigate the advantage of Hybrid QEC over QEC under certain conditions.