Certified Quantum Random Numbers from Untrusted Light

A remarkable aspect of quantum theory is that certain measurement outcomes are entirely unpredictable to all possible observers. Such quantum events can be harnessed to generate numbers whose randomness is asserted based upon the underlying physical processes. We formally introduce, design and experimentally demonstrate an ultrafast optical quantum random number generator that uses a totally untrusted photonic source. While considering completely general quantum attacks, we certify and generate in real-time random numbers at a rate of 8.05 Gb/s with a rigorous security parameter of 10^(−10). Our security proof is entirely composable, thereby allowing the generated randomness to be utilised for arbitrary applications in cryptography and beyond. To our knowledge, this represents the fastest composably secure source of quantum random numbers ever reported

On motion in dynamic magnetic resonance imaging: Applications in cardiac function and abdominal diffusion

La imagen por resonancia magnética (MRI), hoy en día, representa una potente herramienta para el diagnóstico clínico debido a su flexibilidad y sensibilidad a un amplio rango de propiedades del tejido. Sus principales ventajas son su sobresaliente versatilidad y su capacidad para proporcionar alto contraste entre tejidos blandos. Gracias a esa versatilidad, la MRI se puede emplear para observar diferentes fenómenos físicos dentro del cuerpo humano combinando distintos tipos de pulsos dentro de la secuencia. Esto ha permitido crear distintas modalidades con múltiples aplicaciones tanto biológicas como clínicas. La adquisición de MR es, sin embargo, un proceso lento, lo que conlleva una solución de compromiso entre resolución y tiempo de adquisición (Lima da Cruz, 2016; Royuela-del Val, 2017). Debido a esto, la presencia de movimiento fisiológico durante la adquisición puede conllevar una grave degradación de la calidad de imagen, así como un incremento del tiempo de adquisición, aumentando así tambien la incomodidad del paciente. Esta limitación práctica representa un gran obstáculo para la viabilidad clínica de la MRI. En esta Tesis Doctoral se abordan dos problemas de interés en el campo de la MRI en los que el movimiento fisiológico tiene un papel protagonista. Éstos son, por un lado, la estimación robusta de parámetros de rotación y esfuerzo miocárdico a partir de imágenes de MR-Tagging dinámica para el diagnóstico y clasificación de cardiomiopatías y, por otro, la reconstrucción de mapas del coeficiente de difusión aparente (ADC) a alta resolución y con alta relación señal a ruido (SNR) a partir de adquisiciones de imagen ponderada en difusión (DWI) multiparamétrica en el hígado.Departamento de Teoría de la Señal y Comunicaciones e Ingeniería TelemáticaDoctorado en Tecnologías de la Información y las Telecomunicacione

Robust Network Topology Inference and Processing of Graph Signals

The abundance of large and heterogeneous systems is rendering contemporary data more pervasive, intricate, and with a non-regular structure. With classical techniques facing troubles to deal with the irregular (non-Euclidean) domain where the signals are defined, a popular approach at the heart of graph signal processing (GSP) is to: (i) represent the underlying support via a graph and (ii) exploit the topology of this graph to process the signals at hand. In addition to the irregular structure of the signals, another critical limitation is that the observed data is prone to the presence of perturbations, which, in the context of GSP, may affect not only the observed signals but also the topology of the supporting graph. Ignoring the presence of perturbations, along with the couplings between the errors in the signal and the errors in their support, can drastically hinder estimation performance. While many GSP works have looked at the presence of perturbations in the signals, much fewer have looked at the presence of perturbations in the graph, and almost none at their joint effect. While this is not surprising (GSP is a relatively new field), we expect this to change in the upcoming years. Motivated by the previous discussion, the goal of this thesis is to advance toward a robust GSP paradigm where the algorithms are carefully designed to incorporate the influence of perturbations in the graph signals, the graph support, and both. To do so, we consider different types of perturbations, evaluate their disruptive impact on fundamental GSP tasks, and design robust algorithms to address them.Comment: Dissertatio

Exciting with quantum light

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica de la Materia Condensada. Fecha de lectura: 22-11-2019A two-level system—the idealization of an atom with only two energy levels—is the most fundamental quantum object. As such, it has long been at the forefront of the research in Quantum Optics: its emission spectrum is simply a Lorentzian distribution, and the light it produces is the most quantum that can be. The temporal distribution of the photon emission displays a perfect antibunching, meaning that such a system will never emit two (or more) photons simultaneously, which is consistent with the intuition that the two-level system can only sustain a single excitation at any given time. Although these two properties have been known for decades, it was not until the advent of the Theory of Frequency-filtered and Time-resolved Correlations that it was observed that the perfect antibunching is not the end of the story: the correlations between photons possess an underlying structure, which is unveiled when one retains the information about the color of the photons. This is a consequence of the Heisenberg uncertainty principle: measuring perfect antibunching implies an absolute knowledge about the time at which the photons have been emitted, which in turn implies an absolute uncertainty on their energy. Thus, keeping some information about the frequency of the emitted photons affects the correlations between them. This means that a two-level system can be turned into a versatile source of quantum light, providing light with a large breadth of correlation types well beyond simply antibunching. Furthermore, when the two-level system is driven coherently in the so-called Mollow regime (in which the two-level system becomes dressed by the laser and the emission line is split into three), the correlations blossom: one can find every type of statistics—from antibunching to super-bunching—provided that one measures the photons emitted at the adequate frequency window of the triplet. In fact, the process of filtering the emission at the frequencies corresponding to N-photon transitions is the idea behind the Bundler, a source of light whose emission is always in bundles of exactly N photons. The versatility of the correlations decking the emitted light motivates the topic of this Dissertation, in which I focus on the theoretical study of the behaviour that arises when physical systems are driven with quantum light, i.e., with light that cannot be described through the classical theory of electromagnetism. As the canon of excitation used in the literature is restricted to classical sources, namely lasers and thermal reservoirs, our description starts with the most fundamental objects that can be considered as the optical targets: a harmonic oscillator (which represents the field for non-interacting bosonic particles) and a two-level system (which in turn represents the field for fermionic particles). We describe which regions of the Harmonic oscillator’s Hilbert space can be accessed by driving the harmonic oscillator with the light emitted by a two-level system, i.e., which quantum steady states can be realized. Analogously, we find that the quality of the single-photon emission from a two-level system can be enhanced when it is driven by quantum light. Once the advantages of using quantum, rather than classical, sources of light are demonstrated with the fundamental optical targets, we turn to the quantum excitation of more involved systems, such as the strong coupling between a harmonic oscillator and either a two-level system—whose description is made through the Jaynes-Cummings model—or a nonlinear harmonic oscillator—which can be realized in systems of, e.g., exciton-polaritons. Here we find that the statistical versatility of the light emitted by the Mollow triplet allows to perform Quantum Spectroscopy on these systems, thus gaining knowledge of its internal structure and dynamics, and in particular to probe their interactions with the least possible amount of particles: two. In the process of exciting with quantum light, we are called to further examine the source itself. In fact, there is even the need to revisit the concept of a single-photon source, for which we propose more robust criterion than g(2). We also turn to toy-models of the Bundler so as to use it effectively as an optical source. We can then xix study the advantages that one gets and shortcomings that one faces when using this source of light to drive all the systems considered on excitation with the emission of a two-level system. Finally, we go from the continuous to the pulsed regime of excitation, which is of higher interest for applications and comes with its own set of fundamental questions

Mean square solutions of random linear models and computation of their probability density function

[EN] This thesis concerns the analysis of differential equations with uncertain input parameters, in the form of random variables or stochastic processes with any type of probability distributions. In modeling, the input coefficients are set from experimental data, which often involve uncertainties from measurement errors. Moreover, the behavior of the physical phenomenon under study does not follow strict deterministic laws. It is thus more realistic to consider mathematical models with randomness in their formulation. The solution, considered in the sample-path or the mean square sense, is a smooth stochastic process, whose uncertainty has to be quantified. Uncertainty quantification is usually performed by computing the main statistics (expectation and variance) and, if possible, the probability density function. In this dissertation, we study random linear models, based on ordinary differential equations with and without delay and on partial differential equations. The linear structure of the models makes it possible to seek for certain probabilistic solutions and even approximate their probability density functions, which is a difficult goal in general. A very important part of the dissertation is devoted to random second-order linear differential equations, where the coefficients of the equation are stochastic processes and the initial conditions are random variables. The study of this class of differential equations in the random setting is mainly motivated because of their important role in Mathematical Physics. We start by solving the randomized Legendre differential equation in the mean square sense, which allows the approximation of the expectation and the variance of the stochastic solution. The methodology is extended to general random second-order linear differential equations with analytic (expressible as random power series) coefficients, by means of the so-called Fröbenius method. A comparative case study is performed with spectral methods based on polynomial chaos expansions. On the other hand, the Fröbenius method together with Monte Carlo simulation are used to approximate the probability density function of the solution. Several variance reduction methods based on quadrature rules and multilevel strategies are proposed to speed up the Monte Carlo procedure. The last part on random second-order linear differential equations is devoted to a random diffusion-reaction Poisson-type problem, where the probability density function is approximated using a finite difference numerical scheme. The thesis also studies random ordinary differential equations with discrete constant delay. We study the linear autonomous case, when the coefficient of the non-delay component and the parameter of the delay term are both random variables while the initial condition is a stochastic process. It is proved that the deterministic solution constructed with the method of steps that involves the delayed exponential function is a probabilistic solution in the Lebesgue sense. Finally, the last chapter is devoted to the linear advection partial differential equation, subject to stochastic velocity field and initial condition. We solve the equation in the mean square sense and provide new expressions for the probability density function of the solution, even in the non-Gaussian velocity case.[ES] Esta tesis trata el análisis de ecuaciones diferenciales con parámetros de entrada aleatorios, en la forma de variables aleatorias o procesos estocásticos con cualquier tipo de distribución de probabilidad. En modelización, los coeficientes de entrada se fijan a partir de datos experimentales, los cuales suelen acarrear incertidumbre por los errores de medición. Además, el comportamiento del fenómeno físico bajo estudio no sigue patrones estrictamente deterministas. Es por tanto más realista trabajar con modelos matemáticos con aleatoriedad en su formulación. La solución, considerada en el sentido de caminos aleatorios o en el sentido de media cuadrática, es un proceso estocástico suave, cuya incertidumbre se tiene que cuantificar. La cuantificación de la incertidumbre es a menudo llevada a cabo calculando los principales estadísticos (esperanza y varianza) y, si es posible, la función de densidad de probabilidad. En este trabajo, estudiamos modelos aleatorios lineales, basados en ecuaciones diferenciales ordinarias con y sin retardo, y en ecuaciones en derivadas parciales. La estructura lineal de los modelos nos permite buscar ciertas soluciones probabilísticas e incluso aproximar su función de densidad de probabilidad, lo cual es un objetivo complicado en general. Una parte muy importante de la disertación se dedica a las ecuaciones diferenciales lineales de segundo orden aleatorias, donde los coeficientes de la ecuación son procesos estocásticos y las condiciones iniciales son variables aleatorias. El estudio de esta clase de ecuaciones diferenciales en el contexto aleatorio está motivado principalmente por su importante papel en la Física Matemática. Empezamos resolviendo la ecuación diferencial de Legendre aleatorizada en el sentido de media cuadrática, lo que permite la aproximación de la esperanza y la varianza de la solución estocástica. La metodología se extiende al caso general de ecuaciones diferenciales lineales de segundo orden aleatorias con coeficientes analíticos (expresables como series de potencias), mediante el conocido método de Fröbenius. Se lleva a cabo un estudio comparativo con métodos espectrales basados en expansiones de caos polinomial. Por otro lado, el método de Fröbenius junto con la simulación de Monte Carlo se utilizan para aproximar la función de densidad de probabilidad de la solución. Para acelerar el procedimiento de Monte Carlo, se proponen varios métodos de reducción de la varianza basados en reglas de cuadratura y estrategias multinivel. La última parte sobre ecuaciones diferenciales lineales de segundo orden aleatorias estudia un problema aleatorio de tipo Poisson de difusión-reacción, en el que la función de densidad de probabilidad es aproximada mediante un esquema numérico de diferencias finitas. En la tesis también se tratan ecuaciones diferenciales ordinarias aleatorias con retardo discreto y constante. Estudiamos el caso lineal y autónomo, cuando el coeficiente de la componente no retardada i el parámetro del término retardado son ambos variables aleatorias mientras que la condición inicial es un proceso estocástico. Se demuestra que la solución determinista construida con el método de los pasos y que involucra la función exponencial retardada es una solución probabilística en el sentido de Lebesgue. Finalmente, el último capítulo lo dedicamos a la ecuación en derivadas parciales lineal de advección, sujeta a velocidad y condición inicial estocásticas. Resolvemos la ecuación en el sentido de media cuadrática y damos nuevas expresiones para la función de densidad de probabilidad de la solución, incluso en el caso de velocidad no Gaussiana.[CA] Aquesta tesi tracta l'anàlisi d'equacions diferencials amb paràmetres d'entrada aleatoris, en la forma de variables aleatòries o processos estocàstics amb qualsevol mena de distribució de probabilitat. En modelització, els coeficients d'entrada són fixats a partir de dades experimentals, les quals solen comportar incertesa pels errors de mesurament. A més a més, el comportament del fenomen físic sota estudi no segueix patrons estrictament deterministes. És per tant més realista treballar amb models matemàtics amb aleatorietat en la seua formulació. La solució, considerada en el sentit de camins aleatoris o en el sentit de mitjana quadràtica, és un procés estocàstic suau, la incertesa del qual s'ha de quantificar. La quantificació de la incertesa és sovint duta a terme calculant els principals estadístics (esperança i variància) i, si es pot, la funció de densitat de probabilitat. En aquest treball, estudiem models aleatoris lineals, basats en equacions diferencials ordinàries amb retard i sense, i en equacions en derivades parcials. L'estructura lineal dels models ens fa possible cercar certes solucions probabilístiques i inclús aproximar la seua funció de densitat de probabilitat, el qual és un objectiu complicat en general. Una part molt important de la dissertació es dedica a les equacions diferencials lineals de segon ordre aleatòries, on els coeficients de l'equació són processos estocàstics i les condicions inicials són variables aleatòries. L'estudi d'aquesta classe d'equacions diferencials en el context aleatori està motivat principalment pel seu important paper en Física Matemàtica. Comencem resolent l'equació diferencial de Legendre aleatoritzada en el sentit de mitjana quadràtica, el que permet l'aproximació de l'esperança i la variància de la solució estocàstica. La metodologia s'estén al cas general d'equacions diferencials lineals de segon ordre aleatòries amb coeficients analítics (expressables com a sèries de potències), per mitjà del conegut mètode de Fröbenius. Es duu a terme un estudi comparatiu amb mètodes espectrals basats en expansions de caos polinomial. Per altra banda, el mètode de Fröbenius juntament amb la simulació de Monte Carlo són emprats per a aproximar la funció de densitat de probabilitat de la solució. Per a accelerar el procediment de Monte Carlo, es proposen diversos mètodes de reducció de la variància basats en regles de quadratura i estratègies multinivell. L'última part sobre equacions diferencials lineals de segon ordre aleatòries estudia un problema aleatori de tipus Poisson de difusió-reacció, en què la funció de densitat de probabilitat és aproximada mitjançant un esquema numèric de diferències finites. En la tesi també es tracten equacions diferencials ordinàries aleatòries amb retard discret i constant. Estudiem el cas lineal i autònom, quan el coeficient del component no retardat i el paràmetre del terme retardat són ambdós variables aleatòries mentre que la condició inicial és un procés estocàstic. Es prova que la solució determinista construïda amb el mètode dels passos i que involucra la funció exponencial retardada és una solució probabilística en el sentit de Lebesgue. Finalment, el darrer capítol el dediquem a l'equació en derivades parcials lineal d'advecció, subjecta a velocitat i condició inicial estocàstiques. Resolem l'equació en el sentit de mitjana quadràtica i donem noves expressions per a la funció de densitat de probabilitat de la solució, inclús en el cas de velocitat no Gaussiana.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P. I acknowledge the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Jornet Sanz, M. (2020). Mean square solutions of random linear models and computation of their probability density function [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138394TESI

The relationship between analogy and categorisation in cognition

This central topic of this thesis is the relationship between categorisation and analogy in cognition. Questions of what a straightforward representation of a concept or category is, and following from that how extra-categorical associations such as analogy and metaphor are possible are central to our understanding of human reasoning and comprehension. However, despite the intimate linkage between the two, the trend in cognitive science has been to treat analogy and categorisation as separable, distinctive phenomena that can be studied in isolation from one another. This strategy has proved remarkably effective when it comes to the cognitive modelling of extracategorical associations. A number of compelling and detailed models of analogy process exist, and there is widespread agreement amongst researchers studying analogy as to what the key cognitive processes that determine analogies are.However, these models of analogy tend to assume some kind of fully specified category processing module which governs and determines ordinary, straightforward conceptual mappings. Indeed, this assumption is required in order to talk about analogy and metaphor in the first place: few theorists actually define analogy and metaphor per se, but all agree that analogical and metaphoric judgements can be defined in contrast to ordinary categorisation judgements.This thesis reviews these models of analogy, and evidence for them, before conducting a detailed exploration of categorisation in relation to analogy. A theoretical and empirical review is presented in order to show that the straightforward notion of categorisation that underpins the distinctive phenomena approach to the study of analogy and categorisation is more apparent than real. Whilst intuitively, analogy and categorisation might feel like different things which can be contrasted with one another, from a cognitive processing point of view, this thesis argues that such a distinction may not survive a detailed scientific examination.A series of empirical studies are presented in order to further explore the 'no distinction' hypothesis. Following from these, further studies examine the question of whether models of analogical processing have progressed as far as they can in artificial isolation from categorisation, a process in which the processes that are normally deemed 'analogical' appear to play a vital role.The conclusion drawn in this thesis is that the analogy / categorisation division, as currently formulated, cannot survive detailed scientific examination. It is argued that despite the benefits that the previous study of these phenomena in isolation have brought in the past, future progress, especially in the development of cognitive models of analogy, is dependent on a more unified approach

Data-Driven Recommender Systems: Sequences of recommendations

This document is about some scalable and reliable methods for recommender systems from a machine learner point of view. In particular it adresses some difficulties from the non stationary case

Homotopy Theory of Monoids and Group Completion

This thesis presents several complete and partial models for the homotopy theory of monoids and the derived functor of group completion. We show that there is a simplicial model structure on the category of reduced simplicial sets that is Quillen equivalent to the Quillen model structure of simplicial monoids. Using this Quillen equivalence we recover the fact that the derived functor of group completion is isomorphic to the homotopy type of loops on the classifying space of a monoid. We use the Street nerve to show that the derived functor of group completion of monoids in the category of ω-groupoids for the Gray tensor product is isomorphic to group completion for simplicial monoids in low degrees. Finally we exploit the connection of ω-groupoids with the theory of rewriting for presentations of monoids to calculate the second homotopy group of the classifying space BM of a monoid M in terms of a chosen presentation by generators and relations.Department of Pure Mathematics and Mathematical Statistics Cambridge Commonwealth Trust Natural Sciences and Engineering Research Council of Canad