Quantum State Estimation and Symmetric Informationally Complete POMs

Abstract

This thesis studies two basic topics in quantum information science: quantum state estimation and symmetric informationally complete probability operator measurements (SIC POMs).

Part I of this thesis focuses on state estimation of finite-dimensional quantum systems in the large-sample scenario. Four natural settings are investigated in the order of sophistication levels: independent and identical measurements with linear reconstruction, as well as optimal reconstruction, adaptive measurements, and collective measurements. We present an overview of the optimal estimation strategies and tomographic efficiencies under the four settings with respect to typical figures of merit.

Part II of this thesis investigates symmetry properties of SIC POMs from two complementary perspectives, namely, group-theoretic approach and graph-theoretic approach. We determine the symmetry groups of all SIC POMs appearing in the literature and establish complete equivalence relations among them. Our study indicates that, except for Hoggar lines, all SIC POMs known so far are covariant with respect to the Heisenberg--Weyl groups.