Yadsan-Appleby, H; (2013) Gaussian and covariant processes in discrete and continuous variable quantum information. Doctoral thesis , UCL (University College London).

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Abstract

Quantum information theory has attracted much interest in the last decade. The cause of this interest is twofold: the exciting applications that the theory promises, such as the realization of quantum computers, but also the possibility that perhaps the theory will enable us to solve the mysteries of quantum physics. In this thesis we touched a wide variety of topics with the modest motivation that perhaps, at the very least, one could get a little more insight into the conceptual problems. Our motivation led us to carry out the work presented in this thesis. We explore entanglement properties of light in the context of quantum memories. Quantum memories are set to be a crucial component of future quantum computers. In the short and medium term, the development of e ective quantum memories would pave the way for the implementation of a variety of quantum information protocols. For the applications it is important to be able to store entanglement. In this thesis we investigate the storage of two mode Gaussian states of light in a QND feedback quantum memory and we examine the question whether it is better to store the state already entangled or whether is better to store a squeezed state which is only entangled after storage. We then turn to a study of some aspects of the theory of SIC-POVMs (Symmetric Informationally Complete Positive Operator Valued Measures). SIC-POVMs potentially have numerous application in quantum information. They have been constructed mathematically in every dimension 67. But it remains an open question whether they can be constructed in every nite dimension. In this thesis we describe an analogy between coherent states of a continuous variables systems and SIC-POVMs in a discrete system. We then go on to examine the Galois group of the extension eld generated by the components of the SIC-POVM ducial vector. We prove a number of theorems about this group. We then go on to actually calculate the group for a SIC-POVM in dimension 6 and show that it has a number of interesting properties. We speculate that this line of research may make a useful contribution to an eventual proof of the existence of SIC-POVMs. Finally we investigate quantum communication via spin chains. One of the key requirements for a functioning quantum information processor is the ability to transport quantum information from one location to another. Spin chains are a tool which might be used for this purpose. There have been many proposals recently which showed that under fairly general conditions spin chains communicate quantum information with arbitrarily high delity. However, so far there have not been many proposals addressing the problem of communicating as much quantum information as possible. In this thesis we address this problem and describe a method which achieves a high transmission rate for long spin chains.

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