Universal resources for quantum information processing over continuous variables

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We know that classical systems can compute, and we exploit this everyday in our modern electrical devices where classical bits of information are routinely processed. A similar computation can happen at the quantum level as well: electrons, photons, and quantum systems in general can store and process quantum bits (qubits) of information. The extraordinary fact is that quantum systems can compute in an unparalleled way, much better than their classical counterpart, with a consequent revolutionary impact for our technologies. To make this promise a reality, it is necessary to identify controllable physical systems able to support the processing of quantum information. Most of the concepts of quantum information were originally developed for finite dimensional systems (qubits). However it was soon realised that a valid and promising alternative is offered also by infinite dimensional systems -- continuous variables in jargon -- the most familiar examples being position and momentum of a quantised harmonic oscillator. This Ph.D. programme will deal with the latter approach, exploring the possibilities offered by continuous-variable systems for quantum information processing.Fully-fledged (universal) quantum information processing over continuous variables is achievable with so-called linear operations only provided that a single non-linear operation is also available. Whereas linear operations have been both demonstrated experimentally and fully characterised theoretically, much less has been achieved for what concerns the essential non-linear operations. In this context, the Ph.D. programme will aim at developing a thorough theoretical understanding of the weak non-linear operations available experimentally nowadays and in the near future, and in particular how they can be used for quantum information technologies over continuous-variable systems.The project will comprise three parts. The focus of the first part will be the characterisation of non-linearities using the innovative theoretical approach of quantum resource theories. The latter is a powerful mathematical framework that has been recently applied to a variety of contexts in quantum information, including recently continuous variables. In the second part of the programme, the physical settings in which non-linearities are starting to become technologically achievable (such as superconducting circuits for quantum optics and opto-mechanics) will be investigated with the aim of proposing actual implementations of the findings of the first part of the project. The third part concerns applications of non-linearities to the emergent field of quantum machine learning, with the aim of developing strategies to build quantum devices (in particular, quantum multi-mode systems of light) that mimic the role of classical neural networks in the standard machine-learning algorithms -- used for example in the context of image recognition. The student will be fully involved in the development of all the parts of this project, starting from the development of a background theoretical knowledge about the methods of quantum resource theories to the development of continuous-variable approaches to quantum machine-learning. The findings of this project will contribute to the research on the EPSRC theme of quantum technologies, with specific focus on quantum computing and quantum optics.