Probing one-dimensional quantum systems through their entanglement structure

Full knowledge of the entanglement properties of quantum systems can be used to identify different phases in condensed matter. Quantum correlations serve as a fingerprint for universal behaviours, leading to the discovery of new phases and new tools for probing them. In this thesis we use quantum correlations, as witnessed by the entanglement spectrum of a bipartitioned state, to probe the phases and behaviours of various one-dimensional quantum systems. In an era when novel quantum technologies are at the forefront of research it is important to find new models and new methods that may be applicable to the field. This thesis is a composition of two main works. The first is a study of a topological phase with non-local couplings, where we find that protected midgap states are split from zero energy whilst retaining their topological properties. The second aims to quantify the applicability of a known approximate method through the optimality of its entanglement spectrum. We determine bounds that confirm regions of applicability and suggest a new model that is by construction always optimal.