On trying to describe quantum mechanics to a non-physicist, one often receives the response 'What does that really mean?' or alternatively 'What use is that?' Although these questions may seem naiVe, their answers are neither obvious nor even fully understood. In this thesis we present research aiming to help build answers to these questions. The first part of the thesis tries to shed some light on 'what quantum mechanics really means' by turning the question on its head and asking 'does quantum mechanics mean anything real?' The foundational issue of whether quantum mechanics is 'compatible with a realistic picture of our universe has puzzled some ofthe greatest physicists for over a hundred years. The approach we adopt is to begin by assuming a priori that objects in our universe can, at some level, be described by definite real states, and then ask what constraints experimental evidence (in agreement with quantum mechanics) will place on any such realistic theory. Regardless of the success of such a realistic description of our universe, our motivation for this 'reductio ad absurdum' approach is that it allows us to extract clues about how any successful description of our universe must fail to be realistic in a purely classical sense. The aim is that this will hint towards the key properties that must distinguishany successful description of our universe from our classical way of thinking. In order to quantitatively pursue this approach we build a generalized extension ofthe ontological model formalism, which allows one to compare and contrast a wide set ofrealistic theories within a single framework. We show how this formalism can clarify existing constraints placed on realism by quantum mechanics, such as non-locality, and then use it to derive a new requirement ofrealistic theories called deficiency. This is closely related to the property of contextuality, which was shown by Kochen and Specker to be necessary in any realistic interpretation of quantum mechanics. After having discussed the possibilities for what quantum behaviour could mean, the second part of this thesis considers how one might be able to make practical use ofthis behaviour, regardless ofhow one interprets it. The fields of quantum computation and quantum information offer very promising applications ofquantum mechanics. However, despite the promise it holds, the transmission and manipulation ofinformation solely with quantum systems still faces challenges which must be overcome before the technology can reach fruition. We consider the problem ofhow, within a completely quantum mechanical architecture, one might use quantum information to provide reference frames for information transfers. Specifically, we discuss how one might produce quantum mechanical reference frames necessary in crucial quantum information tasks such as violating Bell inequalities.

Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum mechanics has changed in an equal dramatic manner our understanding of information processing and computation. On one hand, the fundamental properties of quantum systems can be harnessed to transmit, store, and manipulate information in a much more efficient and secure way than possible in the realm of classical physics. On the other hand, the development of systematic procedures to manipulate systems of a large number of particles in the quantum regime, crucial to the implementation of quantum based information processing, has triggered new possibilities in the exploration of quantum many-body physics and related areas. In this thesis, we present new results relevant to two important problems in quantum information science: the development of a theory of entanglement, intrinsically quantum correlations of key importance in quantum information theory, and the exploration of the use of controlled quantum systems to the computation and simulation of quantum many-body phenomena. In the first part we introduce a new approach to the study of entanglement by considering its manipulation under operations not capable of generating entanglement. In this setting we show how the landscape of entanglement conversion is reduced to the simplest situation possible: one unique measure completely specifying which transformations are achievable. This framework has remarkable connections with the foundations of thermodynamics, which we present and explore. On the way to establish our main result, we develop new techniques that are of interest on their own. First, we extend quantum Stein's Lemma, characterizing optimal rates in state discrimination, to the case where the alternative hypothesis might vary over particular sets of possibly correlated (non-LLd) states. Second, we show how recent advances in quantum de Finetti type theorems can be employed to decide when the entanglement contained in non-LLd. sequences of states is distillable by local operations and classical communication. In the second part we discuss the usefulness of a quantum computer to the determination of properties of many-body systems. Our first result is a new quantum procedure, based on the phase estimation quantum algorithm, to calculate additive approximations to partition functions and spectrum densities of quantum local Hamiltonians. We give convincing evidence that quantum computation is superior to classical in solving both problems by showing that they are complete for the class of problems efficiently solved in the one-c1ean-qubit model of quantum computation, which is believe to contain classically hard problems. We then present a negative result on the usefulness of quantum computers and prove that the determination of the ground state energy of local quantum Hamiltonians, with the promise that the gap is larger than an inverse polynomial in the number of sites, is hard for the class QCMA, which is believed to contain intractable problems even for quantum computation. In the third and last part, we approach the problem of quantum simulating many-body systems from a more pragmatic point of view. Based on recent experimental developments on cavity quantum electrodynamics, more specifically on the fabrication of arrays of interacting micro-cavities and on their coupling to atomic-like structures in several physical set-ups, we propose and analyse the realization of paradigmatic condensed matter models in such systems, such as the Bose-Hubbard and the anisotropic Heisenberg models. We present· promising properties of such coupled-cavity arrays as simulators of quantum many-body physics, such as the full addressability of individual sites and the access to inhomogeneous models, and discuss the feasibility of an experimental realization with state-of-the-art current technology.

This thesis presents a mathematical analysis of two classes of behaviours which occur in systems of populations: noise-induced bistability and stochastic patterning. Both behaviours have their origins in the intrinsic stochasticity possessed by a population system due to the discreteness of the individuals: the intrinsic noise. In the study of noise-induced bistability, we analyse a system which exhibits switching between two states. These states do not correspond to fixed points of the corresponding system of deterministic equations, but instead are the states at which the system stochasticity is minimal or vanishing. This feature suggests that the mechanism is intrinsically different to the traditional paradigm of bistability, in which a system with two stable fixed points is subject to noise. Through our mathematical analysis we highlight some characteristic properties of the dynamics, suggesting a way to distinguish, in a real system, the presence of noise-induced bistable states from other types of bistability. Stochastic patterning arises when noise acts on a reaction-diffusion system which exhibits pattern formation via an instability of the homogeneous state. If the system is close to the onset of the instability, whilst still in the stable regime, then patterning occurs due to a combination of stochastic agitation and the exponential decay of the underlying stable homogeneous state. We investigate the case of the stochastic travelling waves on both regular lattices and complex networks. In both cases, a complete analytical treatment is provided via the power spectra of fluctuations. The spirit of the thesis is to propose a simple model which is representative of an observed behaviour, and then solve the model analytically. Numerical simulations are used throughout to verify the accurateness of the analytical approximations. Thus the analytical treatments constitute the core of the work and have two purposes. They are explanatory, in the sense that they help to develop intuition about how the noise leads to a certain behaviour. Moreover, they give quantitative understanding, as we provide the explicit expressions for various quantities (stationary distributions, mean times, etc.). In some cases, the formulas that we have obtained do not rely on the details of the model, so that we would expect them to fit experimental data. In other cases this is not so, yet the analytical treatment may give insight on how to attack more realistic models.

The rampant success of quantum theory is the result of applications of the 'new' quantum mechanics of Schrödinger and Heisenberg (1926-7), the Feynman-Schwinger-Tomonaga Quantum Electrodynamics (1946-51), the electro-weak theory of Salaam, Weinberg, and Glashow (1967-9), and Quantum Chromodynamics (1973-); in fact, this success of `the' quantum theory has depended on a continuous stream of brilliant and quite disparate mathematical formulations. In this carefully concealed ferment there lie plenty of unresolved difficulties, simply because in churning out fabulously accurate calculational tools there has been no sensible explanation of all that is going on. It is even argued that such an understanding is nothing to do with physics. A long-standing and famous illustration of this is the paradoxical thought-experiment of Einstein, Podolsky and Rosen (1935). Fundamental to all quantum theories, and also their paradoxes, is the location of sub-microscopic objects; or, rather, that the specification of such a location is fraught with mathematical inconsistency. This project encompasses a detailed, critical survey of the tangled history of Position within quantum theories. The first step is to show that, contrary to appearances, canonical quantum mechanics has only a vague notion of locality. After analysing a number of previous attempts at a `relativistic quantum mechanics', two lines of thought are considered in detail. The first is the work of Wan and students, which is shown to be no real improvement on the usual `nonrelativistic' theory. The second is based on an idea of Dirac's - using backwards-in-time light-cones as the hypersurface in space-time. There remain considerable difficulties in the way of producing a consistent scheme here. To keep things nicely stirred up, the author then proposes his own approach - an adaptation of Feynman's QED propagators. This new approach is distinguished from Feynman's since the propagator or Green's function is not obtained by Feynman's rule. The type of equation solved is also different: instead of an initial-value problem, a solution that obeys a time-symmetric causality criterion is found for an inhomogeneous partial differential equation with homogeneous boundary conditions. To make the consideration of locality more precise, some results of Fourier transform theory are presented in a form that is directly applicable. Somewhat away from the main thrust of the thesis, there is also an attempt to explain the manner in which quantum effects disappear as the number of particles increases in such things as experimental realisations of the EPR and de Broglie thought experiments.

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