Microscopic theory for quantum mirages in quantum corrals

Scanning tunneling microscopy permits us to image the Kondo resonance of a single magnetic atom adsorbed on a metallic surface. When the magnetic impurity is placed at the focus of an elliptical quantum corral, a Kondo resonance has been recently observed both on top of the impurity and on top of the focus where no magnetic impurity is present. This projection of the Kondo resonance to a remote point on the surface is referred to as quantum mirage. We present a quantum mechanical theory for the quantum mirage inside an ideal quantum corral and predict that the mirage will occur in corrals with shapes other than elliptical.

Quantum transition state theory and solving a first order master equation for complex reactions numerically

The field of chemical kinetics is an exciting and active field. The prevailing theories make a number of simplifying assumptions that do not always hold in actual cases. Another current problem concerns a development of efficient numerical algorithms for solving the master equations that arise in the description of complex reactions. The objective of the present work is to furnish a completely general and exact theory of reaction rates, in a form reminiscent of transition state theory, valid for all fluid phases and also to develop a computer program that can solve complex reactions by finding the concentrations of all participating substances as a function of time. To do so, the full quantum scattering theory is used for deriving the exact rate law, and then the resulting cumulative reaction probability is put into several equivalent forms that take into account all relativistic effects if applicable, including one that is strongly reminiscent of transition state theory, but includes corrections from scattering theory. Then two programs, one for solving complex reactions, the other for solving first order linear kinetic master equations to solve them, have been developed and tested for simple applications.

The quantum inspired modelling of changing attitudes and self-organising societies

We utilise the well-developed quantum decision models known to the QI community to create a higher order social decision making model. A simple Agent Based Model (ABM) of a society of agents with changing attitudes towards a social issue is presented, where the private attitudes of individuals in the system are represented using a geometric structure inspired by quantum theory. We track the changing attitudes of the members of that society, and their resulting propensities to act, or not, in a given social context. A number of new issues surrounding this "scaling up" of quantum decision theories are discussed, as well as new directions and opportunities.

Developing the quantum probability ranking principle

In this work, we summarise the development of a ranking principle based on quantum probability theory, called the Quantum Probability Ranking Principle (QPRP), and we also provide an overview of the initial experiments performed employing the QPRP. The main difference between the QPRP and the classic Probability Ranking Principle, is that the QPRP implicitly captures the dependencies between documents by means of quantum interference". Subsequently, the optimal ranking of documents is not based solely on documents' probability of relevance but also on the interference with the previously ranked documents. Our research shows that the application of quantum theory to problems within information retrieval can lead to consistently better retrieval effectiveness, while still being simple, elegant and tractable.

On the use of complex numbers in quantum models for information retrieval

Quantum-inspired models have recently attracted increasing attention in Information Retrieval. An intriguing characteristic of the mathematical framework of quantum theory is the presence of complex numbers. However, it is unclear what such numbers could or would actually represent or mean in Information Retrieval. The goal of this paper is to discuss the role of complex numbers within the context of Information Retrieval. First, we introduce how complex numbers are used in quantum probability theory. Then, we examine van Rijsbergen’s proposal of evoking complex valued representations of informations objects. We empirically show that such a representation is unlikely to be effective in practice (confuting its usefulness in Information Retrieval). We then explore alternative proposals which may be more successful at realising the power of complex numbers.

Document ranking with quantum probabilities

In this thesis we investigate the use of quantum probability theory for ranking documents. Quantum probability theory is used to estimate the probability of relevance of a document given a user's query. We posit that quantum probability theory can lead to a better estimation of the probability of a document being relevant to a user's query than the common approach, i. e. the Probability Ranking Principle (PRP), which is based upon Kolmogorovian probability theory. Following our hypothesis, we formulate an analogy between the document retrieval scenario and a physical scenario, that of the double slit experiment. Through the analogy, we propose a novel ranking approach, the quantum probability ranking principle (qPRP). Key to our proposal is the presence of quantum interference. Mathematically, this is the statistical deviation between empirical observations and expected values predicted by the Kolmogorovian rule of additivity of probabilities of disjoint events in configurations such that of the double slit experiment. We propose an interpretation of quantum interference in the document ranking scenario, and examine how quantum interference can be effectively estimated for document retrieval. To validate our proposal and to gain more insights about approaches for document ranking, we (1) analyse PRP, qPRP and other ranking approaches, exposing the assumptions underlying their ranking criteria and formulating the conditions for the optimality of the two ranking principles, (2) empirically compare three ranking principles (i. e. PRP, interactive PRP, and qPRP) and two state-of-the-art ranking strategies in two retrieval scenarios, those of ad-hoc retrieval and diversity retrieval, (3) analytically contrast the ranking criteria of the examined approaches, exposing similarities and differences, (4) study the ranking behaviours of approaches alternative to PRP in terms of the kinematics they impose on relevant documents, i. e. by considering the extent and direction of the movements of relevant documents across the ranking recorded when comparing PRP against its alternatives. Our findings show that the effectiveness of the examined ranking approaches strongly depends upon the evaluation context. In the traditional evaluation context of ad-hoc retrieval, PRP is empirically shown to be better or comparable to alternative ranking approaches. However, when we turn to examine evaluation contexts that account for interdependent document relevance (i. e. when the relevance of a document is assessed also with respect to other retrieved documents, as it is the case in the diversity retrieval scenario) then the use of quantum probability theory and thus of qPRP is shown to improve retrieval and ranking effectiveness over the traditional PRP and alternative ranking strategies, such as Maximal Marginal Relevance, Portfolio theory, and Interactive PRP. This work represents a significant step forward regarding the use of quantum theory in information retrieval. It demonstrates in fact that the application of quantum theory to problems within information retrieval can lead to improvements both in modelling power and retrieval effectiveness, allowing the constructions of models that capture the complexity of information retrieval situations. Furthermore, the thesis opens up a number of lines for future research. These include: (1) investigating estimations and approximations of quantum interference in qPRP; (2) exploiting complex numbers for the representation of documents and queries, and; (3) applying the concepts underlying qPRP to tasks other than document ranking.

The Quantum Probability Ranking Principle for Information Retrieval

While the Probability Ranking Principle for Information Retrieval provides the basis for formal models, it makes a very strong assumption regarding the dependence between documents. However, it has been observed that in real situations this assumption does not always hold. In this paper we propose a reformulation of the Probability Ranking Principle based on quantum theory. Quantum probability theory naturally includes interference effects between events. We posit that this interference captures the dependency between the judgement of document relevance. The outcome is a more sophisticated principle, the Quantum Probability Ranking Principle, that provides a more sensitive ranking which caters for interference/dependence between documents’ relevance.

Quantum Space-Time and Noncommutative Gauge Field Theories

Arguments arising from quantum mechanics and gravitation theory as well as from string theory, indicate that the description of space-time as a continuous manifold is not adequate at very short distances. An important candidate for the description of space-time at such scales is provided by noncommutative space-time where the coordinates are promoted to noncommuting operators. Thus, the study of quantum field theory in noncommutative space-time provides an interesting interface where ordinary field theoretic tools can be used to study the properties of quantum spacetime. The three original publications in this thesis encompass various aspects in the still developing area of noncommutative quantum field theory, ranging from fundamental concepts to model building. One of the key features of noncommutative space-time is the apparent loss of Lorentz invariance that has been addressed in different ways in the literature. One recently developed approach is to eliminate the Lorentz violating effects by integrating over the parameter of noncommutativity. Fundamental properties of such theories are investigated in this thesis. Another issue addressed is model building, which is difficult in the noncommutative setting due to severe restrictions on the possible gauge symmetries imposed by the noncommutativity of the space-time. Possible ways to relieve these restrictions are investigated and applied and a noncommutative version of the Minimal Supersymmetric Standard Model is presented. While putting the results obtained in the three original publications into their proper context, the introductory part of this thesis aims to provide an overview of the present situation in the field.

Coherent quantum Boltzmann equations from cQPA

We reformulate and extend our recently introduced quantum kinetic theory for interacting fermion and scalar fields. Our formalism is based on the coherent quasiparticle approximation (cQPA) where nonlocal coherence information is encoded in new spectral solutions at off-shell momenta. We derive explicit forms for the cQPA propagators in the homogeneous background and show that the collision integrals involving the new coherence propagators need to be resummed to all orders in gradient expansion. We perform this resummation and derive generalized momentum space Feynman rules including coherent propagators and modified vertex rules for a Yukawa interaction. As a result we are able to set up self-consistent quantum Boltzmann equations for both fermion and scalar fields. We present several examples of diagrammatic calculations and numerical applications including a simple toy model for coherent baryogenesis.

Embedding quantum simulators

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Investigation on relationship between charge transfer position and dielectric definition of average energy gap in Eu3+-doped compounds

The dielectric definition of average energy gap E-g of the chemical bond has been calculated quantitatively in Eu3+-doped 30 lanthanide compounds based on the dielectric theory of chemical bond for complex structure crystals. The relationship between the experimental charge transfer (CT) energy of Eu3+ and the corresponding average energy gap E-g has been studied. The results show that the CT energy increases linearly with increasing of the average energy gap E-g. The linear model is obtained. It allows us to predict the CT position of Eu3+-doped lanthanide compounds with knowledge of the crystal structure and index of refraction. Applied to the Ca4GdO(BO3)(3):Eu and Li2Lu5O4(BO3)(3):Eu crystals, the predicted results of CT energies are in good agreement with the experimental values, and it can be concluded that the lowest CT energy in Li2Lu5O4(BO3)(3):Eu originates from the site of Lu1.

Electromagnetic pulse compression and energy localization in quantum plasmas

The evolution of the intensity of a relativistic laser beam propagating through a dense quantum plasma is investigated, by considering different plasma regimes. A cold quantum fluid plasma and then a thermal quantum description(s) is (are) adopted, in comparison with the classical case of reference. Considering a Gaussian beam cross-section, we investigate both the longitudinal compression and lateral/longitudinal localization of the intensity of a finite-radius electromagnetic pulse. By employing a quantum plasma fluid model in combination with Maxwell's equations, we rely on earlier results on the quantum dielectric response, to model beam-plasma interaction. We present an extensive parametric investigation of the dependence of the longitudinal pulse compression mechanism on the electron density in cold quantum plasmas, and also study the role of the Fermi temperature in thermal quantum plasmas. Our numerical results show pulse localization through a series of successive compression cycles, as the pulse propagates through the plasma. A pulse of 100 fs propagating through cold quantum plasma is compressed to a temporal size of approximate to 1.35 attosecond and a spatial size of approximate to 1.08 10(-3) cm. Incorporating Fermi pressure via a thermal quantum plasma model is shown to enhance localization effects. A 100 fs pulse propagating through quantum plasma with a Fermi temperature of 350 K is compressed to a temporal size of approximate to 0.6 attosecond and a spatial size of approximate to 2.4 10(-3) cm. (c) 2010 Elsevier B.V. All rights reserved.

A no-go result on the purification of quantum states

The information encoded in a quantum system is generally spoiled by the influences of its environment, leading to a transition from pure to mixed states. Reducing the mixedness of a state is a fundamental step in the quest for a feasible implementation of quantum technologies. Here we show that it is impossible to transfer part of such mixedness to a trash system without losing some of the initial information. Such loss is lower-bounded by a value determined by the properties of the initial state to purify. We discuss this interesting phenomenon and its consequences for general quantum information theory, linking it to the information theoretical primitive embodied by the quantum state-merging protocol and to the behaviour of general quantum correlations.

From Classical to Quantum Secret Sharing

Dans ce mémoire, nous nous pencherons tout particulièrement sur une primitive cryptographique connue sous le nom de partage de secret. Nous explorerons autant le domaine classique que le domaine quantique de ces primitives, couronnant notre étude par la présentation d’un nouveau protocole de partage de secret quantique nécessitant un nombre minimal de parts quantiques c.-à-d. une seule part quantique par participant. L’ouverture de notre étude se fera par la présentation dans le chapitre préliminaire d’un survol des notions mathématiques sous-jacentes à la théorie de l’information quantique ayant pour but primaire d’établir la notation utilisée dans ce manuscrit, ainsi que la présentation d’un précis des propriétés mathématique de l’état de Greenberger-Horne-Zeilinger (GHZ) fréquemment utilisé dans les domaines quantiques de la cryptographie et des jeux de la communication. Mais, comme nous l’avons mentionné plus haut, c’est le domaine cryptographique qui restera le point focal de cette étude. Dans le second chapitre, nous nous intéresserons à la théorie des codes correcteurs d’erreurs classiques et quantiques qui seront à leur tour d’extrême importances lors de l’introduction de la théorie quantique du partage de secret dans le chapitre suivant. Dans la première partie du troisième chapitre, nous nous concentrerons sur le domaine classique du partage de secret en présentant un cadre théorique général portant sur la construction de ces primitives illustrant tout au long les concepts introduits par des exemples présentés pour leurs intérêts autant historiques que pédagogiques. Ceci préparera le chemin pour notre exposé sur la théorie quantique du partage de secret qui sera le focus de la seconde partie de ce même chapitre. Nous présenterons alors les théorèmes et définitions les plus généraux connus à date portant sur la construction de ces primitives en portant un intérêt particulier au partage quantique à seuil. Nous montrerons le lien étroit entre la théorie quantique des codes correcteurs d’erreurs et celle du partage de secret. Ce lien est si étroit que l’on considère les codes correcteurs d’erreurs quantiques étaient de plus proches analogues aux partages de secrets quantiques que ne leur étaient les codes de partage de secrets classiques. Finalement, nous présenterons un de nos trois résultats parus dans A. Broadbent, P.-R. Chouha, A. Tapp (2009); un protocole sécuritaire et minimal de partage de secret quantique a seuil (les deux autres résultats dont nous traiterons pas ici portent sur la complexité de la communication et sur la simulation classique de l’état de GHZ).

Entanglement analysis of atomic processes and quantum registers

During recent years, quantum information processing and the study of N−qubit quantum systems have attracted a lot of interest, both in theory and experiment. Apart from the promise of performing efficient quantum information protocols, such as quantum key distribution, teleportation or quantum computation, however, these investigations also revealed a great deal of difficulties which still need to be resolved in practise. Quantum information protocols rely on the application of unitary and non–unitary quantum operations that act on a given set of quantum mechanical two-state systems (qubits) to form (entangled) states, in which the information is encoded. The overall system of qubits is often referred to as a quantum register. Today the entanglement in a quantum register is known as the key resource for many protocols of quantum computation and quantum information theory. However, despite the successful demonstration of several protocols, such as teleportation or quantum key distribution, there are still many open questions of how entanglement affects the efficiency of quantum algorithms or how it can be protected against noisy environments. To facilitate the simulation of such N−qubit quantum systems and the analysis of their entanglement properties, we have developed the Feynman program. The program package provides all necessary tools in order to define and to deal with quantum registers, quantum gates and quantum operations. Using an interactive and easily extendible design within the framework of the computer algebra system Maple, the Feynman program is a powerful toolbox not only for teaching the basic and more advanced concepts of quantum information but also for studying their physical realization in the future. To this end, the Feynman program implements a selection of algebraic separability criteria for bipartite and multipartite mixed states as well as the most frequently used entanglement measures from the literature. Additionally, the program supports the work with quantum operations and their associated (Jamiolkowski) dual states. Based on the implementation of several popular decoherence models, we provide tools especially for the quantitative analysis of quantum operations. As an application of the developed tools we further present two case studies in which the entanglement of two atomic processes is investigated. In particular, we have studied the change of the electron-ion spin entanglement in atomic photoionization and the photon-photon polarization entanglement in the two-photon decay of hydrogen. The results show that both processes are, in principle, suitable for the creation and control of entanglement. Apart from process-specific parameters like initial atom polarization, it is mainly the process geometry which offers a simple and effective instrument to adjust the final state entanglement. Finally, for the case of the two-photon decay of hydrogenlike systems, we study the difference between nonlocal quantum correlations, as given by the violation of the Bell inequality and the concurrence as a true entanglement measure.