Entropy Density and Mismatch in High-Rate Scalar Quantization with Rényi Entropy Constraint

Properties of scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($\alpha=0$) and Gray \emph{et al.}'s variable-rate ($\alpha=1$)mismatch results to general values of the entropy order parameter $\alpha

Neuroengineering of Clustering Algorithms

Cluster analysis can be broadly divided into multivariate data visualization, clustering algorithms, and cluster validation. This dissertation contributes neural network-based techniques to perform all three unsupervised learning tasks. Particularly, the first paper provides a comprehensive review on adaptive resonance theory (ART) models for engineering applications and provides context for the four subsequent papers. These papers are devoted to enhancements of ART-based clustering algorithms from (a) a practical perspective by exploiting the visual assessment of cluster tendency (VAT) sorting algorithm as a preprocessor for ART offline training, thus mitigating ordering effects; and (b) an engineering perspective by designing a family of multi-criteria ART models: dual vigilance fuzzy ART and distributed dual vigilance fuzzy ART (both of which are capable of detecting complex cluster structures), merge ART (aggregates partitions and lessens ordering effects in online learning), and cluster validity index vigilance in fuzzy ART (features a robust vigilance parameter selection and alleviates ordering effects in offline learning). The sixth paper consists of enhancements to data visualization using self-organizing maps (SOMs) by depicting in the reduced dimension and topology-preserving SOM grid information-theoretic similarity measures between neighboring neurons. This visualization\u27s parameters are estimated using samples selected via a single-linkage procedure, thereby generating heatmaps that portray more homogeneous within-cluster similarities and crisper between-cluster boundaries. The seventh paper presents incremental cluster validity indices (iCVIs) realized by (a) incorporating existing formulations of online computations for clusters\u27 descriptors, or (b) modifying an existing ART-based model and incrementally updating local density counts between prototypes. Moreover, this last paper provides the first comprehensive comparison of iCVIs in the computational intelligence literature --Abstract, page iv

Exact diagonalization studies of quantum simulators

Understand and tame complex quantum mechanical systems to build quantum technologies is one of the most important scientific endeavour nowadays. In this effort, Atomic, molecular and Optical systems have clearly played a major role in producing proofs of concept of several important applications. Notable examples are Quantum Simulators for difficult problems in other branches of physics i.e. spin systems, disordered systems, etc., and small sized Quantum Computers. In particular, ultracold atomic gases and trapped ion experiments are nowadays at the forefront in the field. This fantastic experimental effort needs to be accompanied by a matching theoretical and numerical one. The main two reasons are: 1) theoretical work is needed to identify suitable regimes where the AMO systems can be used as efficient quantum simulators of important problems in physics and mathematics, 2) thorough numerical work is needed to benchmark the results of the experiments in parameter regions where a solution to the problem can be found with classical devices. In this dissertation, we present several important examples of systems, which can be numerically solved. The technique used, which is common to all the work presented in the dissertation, is exact diagonalization. This technique works solely for systems of a small number of particles and/or a small number of available quantum states. Despite this limitation, one can study a large variety of quantum systems in relevant parameter regimes. A notable advantage is that it allows one to compute not only the ground state of the system but also most of the spectrum and, in some cases, to study dynamics. The dissertation is organized in the following way. First, we provide an introduction, outlining the importance of this technique for quantum simulation and quantum validation and certification. In Chapter 2, we detail the exact diagonalization technique and present an example of use for the phases of the 1D Bose-Hubbard chain. Then in Chapters 3 to 6, we present a number of important uses of exact diagonalization. In Chapter 3, we study the quantum Hall phases, which are found in two-component bosons subjected to artificial gauge fields. In Chapter 4, we turn into dynamical gauge fields, presenting the topological phases which appear in a bosonic system trapped in a small lattice. In Chapter 5, a very different problem is tackled, that of using an ultracold atomic gases to simulate a spin model. Quantum simulation is again the goal of Chapter 6, where we propose a way in which the number-partitioning problem can be solved by means of a quantum simulator made with trapped ions. Finally, in Chapter 7, we collect the main conclusions of the dissertation and provide a brief outlook.Entendre i controlar sistemes complexos regits per la mecànica quàntica per a construir tecnologies quàntiques es un dels reptes mes rellevants de la ciència en l’actualitat. Els sistemes atòmics, moleculars i òptics han jugat clarament un rol capital en aquest esforç, produint proves de concepte per a diverses aplicacions de consideració. Exemples notables en son els simuladors quàntics dissenyats per a resoldre problemes complicats d’altres branques de la física, com ara sistemes d’espins, sistemes desordenats, etc.... i ordinadors quàntics de dimensions reduïdes. En particular, els experiments amb gasos d’àtoms ultrafreds i amb trampes iòniques son la punta de llança del camp en l’actualitat. El fantàstic afany experimental ha d’anar associat amb d’altres teòric i numèric que el corresponguin. Les raons principals son: 1) els estudis teòrics son necessaris per tal d’identificar règims adients en que els sistemes AMO puguin esser emprats com a simuladors quàntics eficients de problemes rellevants de la Física i les Matemàtiques, 2) els treballs numèrics exhaustius son necessaris per a contrastar els resultats dels experiments en regions de paràmetres en que els dispositius clàssics son capaços de trobar solucions. En aquesta tesi, presentem diversos exemples de sistemes rellevants que poden esser resolts numèricament. La tècnica emprada -que es comuna per a tot el treball- es la diagonalització exacta. L’ús d’aquesta tècnica es limitat a sistemes amb nombres baixos partícules i/o pocs estats quàntics accessibles. Malgrat aquesta limitació, es poden estudiar una gran varietat de sistemes quàntics en els règims rellevants dels paràmetres de control. Un avantatge notable es el fet que permet calcular no nomes l’estat de mínima energia del sistema, sinó que també la majoria de l’espectre i, en alguns casos, àdhuc estudiar-ne la dinàmica. La tesi s’organitza tal i com prossegueix. En primer lloc, proveïm una introducció, subratllant la importància d’aquesta tècnica per a la simulació quàntica i la validació quàntica i certificació. En el capítol 2, detallem la tècnica de la diagonalització exacta i presentem un exemple del seu us per a les fases per a una cadena de Bose-Hubbard unidimensional. En els capítols del 3 al 6, presentem alguns usos rellevants de la diagonalització exacta. En el capítol 3, estudiem les fases degudes a l’efecte Hall quàntic en un sistema de dues components de bosons sotmesos a camps de gauge artificials. En el capítol 4, canviem a camps de gauge dinàmics, presentant les fases topològiques que apareixen en un sistema de bosons atrapats en una petita xarxa reticular. En el capítol 5, s’hi tracta un problema ben diferent, el d’emprar gasos d’àtoms ultrafreds per a per a simular un model d’espín. La simulació quàntica es de nou l’objectiu del capítol 6, en que proposem una forma en que el problema de la partició de nombres pot esser resolt per mitja d’un simulador quàntic construït amb trampes iòniques. Finalment, en el capítol 7, recollim les conclusions principals del treball i donem una breu opinió del futur d’aquesta investigació.Entender y controlar sistemas complejos regidos por la mecánica cuántica para construir tecnologías cuánticas es una de los retos científicos más relevantes en la actualidad. Los sistemas atómicos, moleculares y ópticos han jugado claramente un rol capital en este esfuerzo, produciendo pruebas de concepto para diversas aplicaciones de consideración. Notables ejemplos son los simuladores cuánticos diseñados para resolver problemas complicados de otras ramas de la física, como lo son los sistemas de espines, sistemas desordenados, etc.. . . i los ordenadores cuánticos de dimensiones reducidas. En particular, los experimentos con gases de átomos ultrafríos y con trampas iónicas son la punta de lanza del campo en la actualidad. El fantástico empeño experimental tiene que ir asociado a otros teórico y numérico que le correspondan. Las principales razones son: 1) los estudios teóricos son necesarios para identificar regímenes adecuados en que los sistemas AMO puedan ser usados cómo simuladores cuánticos eficientes para problemas relevantes de la Física y las Matemáticas, 2) los trabajos numéricos exhaustivos son necesarios para contrastar los resultados de los experimentos en regiones de parámetros en que los dispositivos clásicos sean capaces de encontrar soluciones. En esta tesis, presentamos diferentes ejemplos de sistemas relevantes que pueden ser resueltos numéricamente. La técnica usada –que es común en todo el trabajo– es la diagonalización exacta. El uso de ésta técnica está restringido a sistemas con números bajos de partículas i/o estados cuánticos accesibles. A pesar de esta limitación, se puede estudiar gran variedad de sistemas cuánticos en los regímenes relevantes de los parámetros de control. Una ventaja notable es que permite calcular no sólo el estado de mínima energía del sistema, sino que también la mayoría del espectro e, en algunos casos, incluso estudiar la dinámica. La tesis se organiza como sigue. En primer lugar, ofrecemos una introducción, subrayando la importancia de esta técnica para la simulación cuántica y la validación cuántica y certificación. En el capítulo 2, detallamos la técnica de la diagonalización exacta y presentamos un ejemplo de su uso para una cadena de Bose-Hubbard unidimensional. En los capítulos del 3 al 6, presentamos algunos usos relevantes de la diagonalización exacta. En el capítulo 3, estudiamos las fases debidas al efecto Hall cuántico en un sistema de dos componentes de bosones sometidos a campos de gauge artificiales. En el capítulo 4, cambiamos hacia campos gauge dinámicos, presentando las fases topológicas que aparecen en un sistema de bosones atrapados en una pequeña malla reticular. En el capítulo 5, se trata un problema bien diferente, el de usar gases de átomos ultrafríos para simular un modelo de espín. La simulación cuántica es de nuevo el objetivo del capítulo 6, en que proponemos una forma en que el problema de la partición de números puede ser resuelta mediante un simulador cuántico construido con trampas iónicas. Finalmente, en el capítulo 7, recogemos las conclusiones principales de los trabajos y damos una breve opinión del futuro de ésta investigaciónPostprint (published version

Computational Intelligence and Complexity Measures for Chaotic Information Processing

This dissertation investigates the application of computational intelligence methods in the analysis of nonlinear chaotic systems in the framework of many known and newly designed complex systems. Parallel comparisons are made between these methods. This provides insight into the difficult challenges facing nonlinear systems characterization and aids in developing a generalized algorithm in computing algorithmic complexity measures, Lyapunov exponents, information dimension and topological entropy. These metrics are implemented to characterize the dynamic patterns of discrete and continuous systems. These metrics make it possible to distinguish order from disorder in these systems. Steps required for computing Lyapunov exponents with a reorthonormalization method and a group theory approach are formalized. Procedures for implementing computational algorithms are designed and numerical results for each system are presented. The advance-time sampling technique is designed to overcome the scarcity of phase space samples and the buffer overflow problem in algorithmic complexity measure estimation in slow dynamics feedback-controlled systems. It is proved analytically and tested numerically that for a quasiperiodic system like a Fibonacci map, complexity grows logarithmically with the evolutionary length of the data block. It is concluded that a normalized algorithmic complexity measure can be used as a system classifier. This quantity turns out to be one for random sequences and a non-zero value less than one for chaotic sequences. For periodic and quasi-periodic responses, as data strings grow their normalized complexity approaches zero, while a faster deceasing rate is observed for periodic responses. Algorithmic complexity analysis is performed on a class of certain rate convolutional encoders. The degree of diffusion in random-like patterns is measured. Simulation evidence indicates that algorithmic complexity associated with a particular class of 1/n-rate code increases with the increase of the encoder constraint length. This occurs in parallel with the increase of error correcting capacity of the decoder. Comparing groups of rate-1/n convolutional encoders, it is observed that as the encoder rate decreases from 1/2 to 1/7, the encoded data sequence manifests smaller algorithmic complexity with a larger free distance value

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Quantum Transport in Mesoscopic Systems

Mesoscopic physics deals with systems larger than single atoms but small enough to retain their quantum properties. The possibility to create and manipulate conductors of the nanometer scale has given birth to a set of phenomena that have revolutionized physics: quantum Hall effects, persistent currents, weak localization, Coulomb blockade, etc. This Special Issue tackles the latest developments in the field. Contributors discuss time-dependent transport, quantum pumping, nanoscale heat engines and motors, molecular junctions, electron–electron correlations in confined systems, quantum thermo-electrics and current fluctuations. The works included herein represent an up-to-date account of exciting research with a broad impact in both fundamental and applied topics