Classically electrons in a three-dimensional solid can change their momentum in all possible directions. However, electrons in semiconductors can be manipulated so that they are constrained to move in lower dimensions. One of the perfect examples of such a system is a semiconductor heterostructure of GaAs/AlGaAs forming a plane of electrons, only a few nanometer thick, at its junction where electrons possessing quantised energy and freedom to change momentum in the plane. Such remarkable ensemble of non-interacting electrons is known as the two-dimensional electron gas (2DEG). The electrons in a 2DEG system are highly mobile and at low temperatures their motion is mainly scattering free due to the reduction in the interaction with lattice vibrations (phonons) and there is little impurity scattering. When the 2D electrons are electrostatically squeezed to form a narrow, 1D channel whose effective size is less than the electron mean free path for scattering then quantum phenomena associated with the electrons becomes resolved. In this situation, the energy of 1D electrons becomes quantised and discrete levels are formed. At a low carrier concentration of electrons, if the potential which is confining the 1D electrons is relaxed then electrons can arrange themselves into a periodic zig- zag manner forming a Wigner Crystal, named after Wigner who first predicted such a phenomenon in metal in 1936. Recently the distortion of a line of electrons into a zig-zag and then into two separate rows of electrons was observed and associated rich spin and charge phases. A very subtle change in confinement can result in two rows emerging from a zig-zag state which indicates that there is a narrow range where wavefunctions separate and form entangled states. Entanglement is a remarkable phenomenon in which a change in state of one electron will introduce a change in state of another. This amazing property forms the basis for quantum information processing with practical consequences related to quantum technologies, which will be investigated in this proposal. Another most important aspect of my Fellowship proposal is investigating the zig-zag regime or relaxed 1D system in search of fractional quantum states in the absence of a magnetic field. In the presence of a large magnetic field the energy of a 2DEG is quantized to form Landau levels which gave rise to two celebrated discoveries of the Integer and fractional quantum Hall effects in 1980 and 1982 respectively. Such unexpected revelations then pose a question whether fractional quantised states in the absence of any magnetic field in any lattice or topological insulators could ever be observed? However, there were no reports of observations of any fractional states without a magnetic field until the recent discovery of fractional charges of e/2 and e/4 arising from the relaxed zig-zag state in a Germanium-based 1D system. The proposal is inspired by this and the recent experimental finding of non-magnetic self-organised fractional quantum states in tradition GaAs based 1D quantum wires, which was completely unanticipated. The research aim is to introduce new insights, and new aspects of quantum physics, by exploiting the interaction effects in low-dimensional semiconductors by manipulating electron wavefunctions in a controllable manner to allow technological exploitation of basic quantum physics. The major challenges to be investigated: spin and charge manipulation, demonstrating electron entanglement and detection, mapping self-organised fractional states and their spin states, controlled manipulation and detection of hybrid fractional states and establishing if they are entangled. This research proposal opens up a new area in the quantum physics of condensed matter with the generation of Non-Abelian fractions which can be used in a Topological Quantum Computation scheme.