Gaussian resource theories and semidefinite programming hierarchies for quantum information

Determining which quantum tasks we can perform with currently available tools and devices is one of the most important goals of quantum information science today. To achieve this requires careful investigation of the capability of current quantum tools as well as development of classical protocols which can assist quantum tasks and amplify their abilities. In this thesis, we approach this problem through two different topics in quantum information theory: Gaussian resource theories and semidefinite programming hierarchies. In the first part of this thesis, we examine the possibility of implementing quantum information processing tasks in the Gaussian platform through the eyes of quantum resource theories. Gaussian states and operations are primary tools for the study of continuous-variable quantum information processing due to their easy accessibility and concise mathematical descriptions, although it has been discovered that they are subject to a number of limitations for advanced quantum information processing tasks. We explore the capability of the Gaussian platform further in the first part of this thesis. Firstly, we investigate whether introducing convex structure to the Gaussian framework can circumvent the known no-go theorem of Gaussian resource distillation. Surprisingly, we find that resource distillation becomes possible — albeit in a limited fashion — when convexity is introduced. Then, we consider the quantum resource theory of Gaussian thermal operations when catalysts are allowed, and examine the abilities of catalytic Gaussian thermal operations by characterising all possible state transformations under them. In the second part of this thesis, we address the problem of characterising quantum cor- relations via semidefinite programming hierarchies. In particular, we focus on characterising quantum correlations of fixed dimension, which is practically relevant to the field of semi- device-independent quantum information processing. Semidefinite programming is a special type of mathematical optimisation, and it is known that some important but difficult problems in quantum information theory admit semidefinite programming relaxations; these include the characterisation of general quantum correlations in the context of non-locality and the distinction of quantum separable states from entangled states. In this second part, we show how to construct a hierarchy of semidefinite programming relaxations for quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. For the proof, we make a connection to a variant of quantum separability problem and employ multipartite quantum de Finetti theorems with linear constraints. Open Access