Cette thèse, constituée de trois papiers de recherche, est organisée autour de la construction d’indicateurs de crises financières dont les signaux sont ensuite utilisés pour l’élaboration de stratégies de trading algorithmique. Le premier papier traite de l’établissement d’un cadre de travail permettant la construction des indicateurs de crises financière. Le pouvoir de prédiction de nos indicateurs est ensuite démontré en utilisant l’un d’eux pour construire une stratégie de type protective-put active qui est capable de faire mieux en termes de performances qu’une stratégie passive ou, la plupart du temps, que de multiples réalisations d’une stratégie aléatoire. Le second papier va plus loin dans l’application de nos indicateurs de crises à la création de stratégies de trading algorithmique en utilisant le signal combiné d’un grand nombre de nos indicateurs pour gouverner la composition d’un portefeuille constitué d’un mélange de cash et de titres d’un ETF répliquant un indice equity comme le SP500. Enfin, dans le troisième papier, nous construisons des indicateurs de crises financières en utilisant une approche complètement différente. En étudiant l’évolution dynamique de la distribution des spreads des composantes d’un indice CDS tel que l’ITRAXXX Europe 125, une bande de Bollinger est construite autour de la fonction de répartition de la distribution empirique des spreads, exprimée sur une base de deux distributions log-normales choisies à l’avance. Le passage par la fonction de répartition empirique de la frontière haute ou de la frontière basse de cette bande de Bollinger est interprétée en termes de risque et permet de produire un signal de trading. / This thesis is constituted of three research papers and is articulated around the construction of financial crisis indicators, which produce signals, which are then applied to devise successful systematic trading strategies. The first paper deals with the establishment of a framework for the construction of our financial crisis indicators. Their predictive power is then demonstrated by using one of them to build an active protective-put strategy, which is able to beat in terms of performance a passive strategy as well as, most of the time, multiple paths of a random strategy. The second paper goes further in the application of our financial crisis indicators to the elaboration of systematic treading strategies by using the aggregated signal produce by many of our indicators to govern a portfolio constituted of a mix of cash and ETF shares, replicating an equity index like the SP500. Finally, in the third paper, we build financial crisis indicators by using a completely different approach. By studying the dynamics of the evolution of the distribution of the spreads of the components of a CDS index like the ITRAXX Europe 125, a Bollinger band is built around the empirical cumulative distribution function of the distribution of the spreads, fitted on a basis constituted of two lognormal distributions, which have been chosen beforehand. The crossing by the empirical cumulative distribution function of either the upper or lower boundary of this Bollinger band is then interpreted in terms of risk and enables us to construct a trading signal.

In the 1880s and early 1890s, the Prague mathematician Eduard Weyr published his important results in matrix theory. His works represented the only significant contribution to matrix theory by Czech mathematicians in many decades that followed. Although Eduard Weyr was one of the few European mathematicians acquainted with matrix theory and working in it at that time, his results did not gain recognition for about a century. Eduard Weyr discovered the Weyr characteristic, which is a dual sequence to the better known Segre characteristic, and also the so-called typical form. This canonical form of a matrix is nowadays called the Weyr canonical form. It is permutationally similar to the commonly used Jordan canonical form of the same matrix and it outperforms the Jordan canonical form in some mathematical situations. The Weyr canonical form has become much better known in the last few years and even a monograph dedicated to this topic was published in 2011.

L'une des stratégies utilisée pour augmenter l'efficacité spectrale (ES) des réseaux cellulaires est de réutiliser la bande de fréquences sur des zones relativement petites. Le problème majeur dans ce cas est un plus grand niveau d'interférence, diminuant l'efficacité énergétique (EE). En plus d'une plus grande largeur de bande, la densification des réseaux (cellules de petite taille ou multi-utilisateur à entrées multiples et sortie unique, MU-EMSO), peut augmenter l'efficacité spectrale par unité de surface (ESuS). La consommation totale d'énergie des réseaux sans fil augmente en raison de la grande quantité de puissance de circuit consommée par les structures de réseau denses, réduisant l'EE. Dans cette thèse, la région EE-SE est caractérisé dans un réseau cellulaire hexagonal en considérant plusieurs facteurs de réutilisation de fréquences (FRF), ainsi que l'effet de masquage. La région EE-ESuS est étudiée avec des processus de Poisson ponctuels (PPP) pour modéliser un réseau MU-EMSO avec un précodeur à rapport signal sur fuite plus bruit (RSFB). Différentes densités de station de base (SB) et nombre d'antennes aux SB avec une consommation d'énergie statique sont considérées.Nous caractérisons d'abord la région EE-SE dans le réseau cellulaire hexagonal pour différentes FRF, avec et sans masquage. Avec le masquage en plus de la perte de propagation, la mesure de coupure ε-EE-ES est proposée pour évaluer les performances. Les courbes EE-ES présentent une grande partie linéaire, due à la consommation de puissance statique, suivie d'une forte diminution de l'EE, puisque le réseau est homogène et limité par les interférences. Les résultats montrent qu'un FRF de 1 pour les régions proches de la SB et des FRF plus élevés dans la région plus proche du bord de la cellule améliorent le point optimal du EE-ES. De plus, un meilleur compromis EE-ES peut être obtenu avec une valeur plus élevée de coupure. En outre, un FRF de 1 est le meilleur choix pour une valeur élevée de coupure en raison d'une réduction du rapport signal sur interférence plus bruit (RSIB).Les précodeurs sont utilisés en liaison descendante des réseaux cellulaires MU-EMSO à accès multiple par division spatiale (AMDS) pour améliorer le RSIB. La géométrie stochastique a été utilisée intensivement pour analyser de tels systèmes complexes. Nous obtenons une expression analytique de l'ESuS en régime asymptotique, c.-à-d. nombre d'antennes et d'utilisateurs infinis, en utilisant des résultats de matrices aléatoires et de géométrie stochastique. Les SBs et les utilisateurs sont modélisés par deux PPP indépendants et le précodage RSFB est utilisé. L'EE est dérivée d'un modèle de consommation de puissance linéaire. Les simulations de Monte Carlo montrent que les expressions analytiques sont précises même pour un nombre faible d'antennes et d'utilisateurs. De plus, les courbes d'EE-ESuS ont une grande partie linéaire avant une forte décroissante de l'EE, comme pour les réseaux hexagonaux. Les résultats montrent également que le précodeur RSFB offre de meilleurs performances que le précodeur forçage à zéro (FZ), qui est typiquement utilisé dans la literature. Les résultats numériques pour le précodeur RSFB montrent que déployer plus de SBs ou d'antennes aux BSs augmente l'ESuS, mais que le gain dépend du rapport des densités SB-utilisateurs et du nombre d'antennes lorsque la densité de l'utilisateur est fixe. L'EE augmente seulement lorsque l'augmentation de l'ESuS est plus importante que l'augmentation de la consommation d'énergie par unité de surface. D'autre part, lorsque la densité d'utilisateur augmente, l'ESuS dans la région limitée par les interférences peut être améliorée en déployant davantage de SB sans sacrifier l'EE et le débit ergodique des utilisateurs. / One of the used strategies to increase the spectral efficiency (SE) of cellular network is to reuse the frequency bandwidth over relatively small areas. The major issue in this case is higher interference, decreasing the energy efficiency (EE). In addition to the higher bandwidth, densification of the networks (e.g. small cells or multi-user multiple input single output, MU-MISO) potentially increases the area spectral efficiency (ASE). The total energy consumption of the wireless networks increases due to the large amount of circuit power consumed by the dense network structures, leading to the decrease of EE. In this thesis, the EE-SE achievable region is characterized in a hexagonal cellular network considering several frequency reuse factors (FRF), as well as shadowing. The EE-ASE region is also studied using Poisson point processes (PPP) to model the MU-MISO network with signal-to-leakage-and-noise ratio (SLNR) precoder. Different base station (BS) densities and different number of BS antennas with static power consumption are considered.The EE-SE region in a hexagonal cellular network for different FRF, both with and without shadowing is first characterized. When shadowing is considered in addition to the path loss, the ε-SE-EE tradeoff is proposed as an outage measure for performance evaluation. The EE-SE curves have a large linear part, due to the static power consumption, followed by a sharp decreasing EE, since the network is homogeneous and interference-limited. The results show that FRF of 1 for regions close to BS and higher FRF for regions closer to the cell edge improve the EE-SE optimal point. Moreover, better EE-SE tradeoff can be achieved with higher outage values. Besides, FRF of 1 is the best choice for very high outage value due to the significant signal-to-interference-plus-noise ratio (SINR) decrease.In downlink, precoders are used in space division multiple access (SDMA) MU-MISO cellular networks to improve the SINR. Stochastic geometry has been intensively used to analyse such a complex system. A closed-form expression for ASE in asymptotic regime, i.e. number of antennas and number of users grow to infinity, has been derived using random matrix theory and stochastic geometry. BSs and users are modeled by two independent PPP and SLNR precoder is used at BS. EE is then derived from a linear power consumption model. Monte Carlo simulations show that the analytical expressions are tight even for moderate number of antennas and users. Moreover, the EE-ASE curves have a large linear part before a sharply decreasing EE, as observed for hexagonal network. The results also show that SLNR outperforms the zero-foring (ZF) precoder, which is typically used in literature. Numerical results for SLNR show that deploying more BS or a large number of BS antennas increase ASE, but the gain depends on the BS-user density ratio and on the number of antennas when user density is fixed. EE increases only when the increase in ASE dominates the increase of the power consumption per unit area. On the other hand, when the user density increases, ASE in interference-limited region can be improved by deploying more BS without sacrificing EE and the ergodic rate of the users.

Três ensembles, respectivamente relacionados com as distribuições Gaussiana, Lognormal e de Levy, são abordados neste trabalho primordialmente do ponto de vista da termodinâmica de seus espectros. Novas expressões para as grandezas termodinâmicas sao encontradas para os ensembles de Stieltjes e de Bertuola-Pato, e a conexão destes com os ensembles Gaussianos e estabelecida. Esta tese também se compromete com a continuação do desenvolvimento e aprimorarão do ensemble generalizado de Bertuola-Pato, estendendo alguns resultados para os ensembles simplifico e unitário generalizados, alem do ortogonal generalizado já introduzido anteriormente por A. C. Bertuola e M. P. Pato. / Three ensembles, related to the Gaussian, the Lognormal and the L´evy distributions respectively, have been studied in this work and were investigated most of all in what concerns their spectral thermodynamics. New expressions for the thermodynamics quantities were found for the Stieltjes and the Bertuola-Pato ensembles, and the connection with the gaussian ensembles is established. This work concerned with the development continuity and with the improvement of Bertuola-Pato generalized ensemble, extending some of the results to the simplectic and unitary generalized ensembles, besides the orthogonal generalized ensemble introduced before by A. C. Bertuola and M. P. Pato.

Orientador: Rogerio Custodio / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-12T23:54:09Z (GMT). No. of bitstreams: 1 Angelotti_WagnerFernandoDelfino_D.pdf: 827045 bytes, checksum: 044a9aecfaec7ac3e14153480c625d9f (MD5) Previous issue date: 2009 / Resumo: Esta tese explorou diferentes objetivos envolvendo o método Monte Carlo Quântico, dos quais se destacam: avaliação do método convencional em propriedades eletrônicas; formalização das relações existentes entre a teoria de matriz densidade e os métodos Monte Carlo Quântico Variacional e de Difusão; estudo da correlação eletrônica com diferentes funções correlacionadas e também através de método misto envolvendo a teoria de perturbação e o Monte Carlo Quântico Variacional; aplicações para átomos do primeiro e segundo período da tabela periódica e moléculas diatômicas. Experimentos computacionais com o método Monte Carlo Quântico e separação de spins foram realizados produzindo excelentes resultados para cálculos de potenciais de ionização sucessivos para átomos, ionização atômica e molecular e construção de curvas de potencial para moléculas simples. Foram ainda obtidas duas formulações analíticas que descrevem exatamente o vínculo formal entre a matriz densidade e o Monte Carlo Quântico. Esta associação proporcionou ótimos resultados para os métodos Variacional e de Difusão, apresentando semelhanças e significativas diferenças quando comparado ao tratamento convencional com respeito à estrutura nodal para cada estado eletrônico estudado. Além disso, a matriz densidade aliada às funções correlacionadas é capaz de recuperar parte da correlação eletrônica e torna possível a correção de funções de onda dentro da associação do Monte Carlo Quântico e teoria de perturbação. / Abstract: This thesis explored different goals involving the quantum Monte Carlo method, of which stand out: assessment of the conventional method in electronic properties; formalization of relations between the density matrix theory and the variational and diffusion quantum Monte Carlo methods; study of the electronic correlation with different correlated functions and also through mixed method involving the perturbation theory and variational quantum Monte Carlo; applications to atoms of the first and second period of the periodic table and diatomic molecules. Computational experiments with the quantum Monte Carlo method and separation of spins were achieved producing excellent results for calculations of successive ionization potentials for atoms, single ionization of atoms and simple molecules and calculation of potential curves for simple molecules. Two analytical formulations were obtained that describes exactly the formal link between the density matrix and quantum Monte Carlo. This association has provided excellent results for variational and diffusion methods, presenting similarities and significant differences when compared to conventional treatment with respect to the nodal structure for each electronic state studied. Furthermore, the density matrix together with correlated wave functions is able to recover part of the electronic correlation and makes possible the correction of the wave functions within the association of quantum Monte Carlo and perturbation theory. / Doutorado / Físico-Química / Doutor em Ciências

At a first glance, artificial neural networks, with engineered learning algorithms and carefully chosen nonlinearities, are nothing like the complicated self-organized spiking neural networks studied by theoretical neuroscientists. Yet, both adapt to their inputs, keep information from the past in their state space and are able of learning, implying that some information processing principles should be common to both. In this thesis we study those principles by incorporating notions of systems theory, statistical physics and graph theory into artificial neural networks and theoretical neuroscience models. % TO DO: What is different in this thesis? -> classical signal processing with complex systems on top The starting point for this thesis is \ac{RC}, a learning paradigm used both in machine learning\cite{jaeger2004harnessing} and in theoretical neuroscience\cite{maass2002real}. A neural network in \ac{RC} consists of two parts, a reservoir – a directed and weighted network of neurons that projects the input time series onto a high dimensional space – and a readout which is trained to read the state of the neurons in the reservoir and combine them linearly to give the desired output. In classical \ac{RC}, the reservoir is randomly initialized and left untrained, which alleviates the training costs in comparison to other recurrent neural networks. However, this lack of training implies that reservoirs are not adapted to specific tasks and thus their performance is often lower than that of other neural networks. Our contribution has been to show how knowledge about a task can be integrated into the reservoir architecture, so that reservoirs can be tailored to specific problems without training. We do this design by identifying two features that are useful for machine learning: the memory of the reservoir and its power spectra. First we show that the correlations between neurons limit the capacity of the reservoir to retain traces of previous inputs, and demonstrate that those correlations are controlled by moduli of the eigenvalues of the adjacency matrix of the reservoir. Second, we prove that when the reservoir resonates at the frequencies that are present on the desired output signal, the performance of the readout increases. Knowing the features of the reservoir dynamics that we need, the next question is how to impose them. The simplest way to design a network with that resonates at a certain frequency is by adding cycles, which act as feedback loops, but this also induces correlations and hence memory modifications. To disentangle the frequencies and the memory design, we studied how the addition of cycles modifies the eigenvalues in the adjacency matrix of the network. Surprisingly, the shape of the eigenvalues is quite beautiful \cite{aceituno2019universal} and can be characterized using random matrix theory tools. Combining this knowledge with our result relating eigenvalues and correlations, we designed an heuristic that tailors reservoirs to specific tasks and showed that it improves upon state of the art \ac{RC} in three different machine learning tasks. Although this idea works in the machine learning version of \ac{RC}, there is one fundamental problem when we try to translate to the world of theoretical neuroscience: the proposed frequency adaptation requires prior knowledge of the task, which might not be plausible in a biological neural network. Therefore the following questions are whether those resonances can emerge by unsupervised learning, and which kind of learning rules would be required. Remarkably, these resonances can be induced by the well-known Spike Time-Dependent Plasticity (STDP) combined with homeostatic mechanisms. We show this by deriving two self-consistent equations: one where the activity of every neuron can be calculated from its synaptic weights and its external inputs and a second one where the synaptic weights can be obtained from the neural activity. By considering spatio-temporal symmetries in our inputs we obtained two families of solutions to those equations where a periodic input is enhanced by the neural network after STDP. This approach shows that periodic and quasiperiodic inputs can induce resonances that agree with the aforementioned \ac{RC} theory. Those results, although rigorous, are expressed on a language of statistical physics and cannot be easily tested or verified in real, scarce data. To make them more accessible to the neuroscience community we showed that latency reduction, a well-known effect of STDP\cite{song2000competitive} which has been experimentally observed \cite{mehta2000experience}, generates neural codes that agree with the self-consistency equations and their solutions. In particular, this analysis shows that metabolic efficiency, synchronization and predictions can emerge from that same phenomena of latency reduction, thus closing the loop with our original machine learning problem. To summarize, this thesis exposes principles of learning recurrent neural networks that are consistent with adaptation in the nervous system and also improve current machine learning methods. This is done by leveraging features of the dynamics of recurrent neural networks such as resonances and correlations in machine learning problems, then imposing the required dynamics into reservoir computing through control theory notions such as feedback loops and spectral analysis. Then we assessed the plausibility of such adaptation in biological networks, deriving solutions from self-organizing processes that are biologically plausible and align with the machine learning prescriptions. Finally, we relate those processes to learning rules in biological neurons, showing how small local adaptations of the spike times can lead to neural codes that are efficient and can be interpreted in machine learning terms.

Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.

In this thesis, we review some applications of random matrix theory in machine learning and theoretical deep learning. More specifically, we review data modelling in the regime of numerous and large dimensional data, a method for estimating covariance matrix distances in the aforementioned regime, as well as an asymptotic analysis of a simple neural network model in the limit where the number of neurons is large and the data is both numerous and large dimensional. We also review some recent research where random matrix models and methods have been applied to Hessian matrices of neural networks with interesting results. As becomes apparent, random matrix theory is a useful tool for various machine learning applications and it is a fruitful field of mathematics toexplore, in particular, in the context of theoretical deep learning. / I denna uppsatsen undersöker vi några tillämpningar av slumpmatristeori inom maskininlärning och teoretisk djupinlärning. Mer specifikt undersöker vi datamodellering i domänet där både datamängden och dimensionen på datan är stor, en metod för att uppskatta avstånd mellan kovariansmatriser i det tidigare nämnda domänet, samt en asymptotisk analys av en enkel neuronnätsmodell i gränsen där antalet neuroner är stort och både datamängden och dimensionen pådatan är stor. Vi undersöker också en del aktuell forskning där slumpmatrismodeller och metoder från slumpmatristeorin har tillämpats på Hessianska matriserför artificiella neuronnätverk med intressanta resultat. Det visar sig att slumpmatristeori är ett användbart verktyg för olika maskininlärningstillämpningaroch är ett område av matematik som är särskilt givande att utforska inom kontexten för teoretisk djupinlärning.

No description available.

It is commonly expected, that the entanglement entropy for eigenstates of quantum chaotic systems can be described by random matrix theory. However, the random matrix predictions account for structureless random states, only. It is unclear, how the subsystem structure of actual bipartite systems influences the entanglement. We investigate the effect of such a structure on the bipartite entanglement for eigenstates of time-periodically kicked Floquet systems. To this end, the expression for the eigenstate entanglement is transferred into a dynamical quantity, which is particularly suited for an evaluation using analytical methods for time evolution. We present three approaches and apply each to an appropriate minimal model. Based on the supersymmetry method, we compute the entanglement of structureless random matrices and thereby establish exact results for the entropy of random matrix eigenstates. The Weingarten calculus is used for computing the entanglement of an inherent bipartite random matrix ensemble. Moreover, based on semiclassical path integrals, we devise a trace formula, which quantifies entanglement of chaotic Floquet systems in terms of classical orbits. We thereby show, that the entanglement of strongly coupled bipartite Floquet systems coincides in the semiclassical limit with the entanglement of structureless random matrices. Several possible generalizations of our methods to autonomous systems and other entropies are discussed.:1. Introduction 2. Fundamentals on bipartite systems and entanglement 2.1. Classical and quantum chaotic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1. Classical mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2. Quantum systems and random matrix theory . . . . . . . . . . . . . . . . . . . 8 2.2. Bipartite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3. Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4. Objective of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3. Random matrix methods for entanglement in bipartite chaotic systems 3.1. Entropy formulation in terms of Green’s functions . . . . . . . . . . . . . . . . . . . . . 23 3.2. Weingarten calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1. Spectral form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2. Inverse participation ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.3. Linear entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3. Linear entropy by the supersymmetry method . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.1. Gaussian integrals and the generating function . . . . . . . . . . . . . . . . . . 44 3.3.2. Supersymmetric integrals and generating function . . . . . . . . . . . . . . . . . 46 3.3.3. Entropy of the CUE case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4. Semiclassical method for entanglement in bipartite chaotic systems 4.1. Path integrals and trace formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.1. Path integral formulation of propagators . . . . . . . . . . . . . . . . . . . . . . 60 4.1.2. Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2. Rescaled path integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.1. Spectral form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.2. Linear entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.3. Order \hbar correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5. Generalizations 5.1. Supersymmetry method for bipartite systems . . . . . . . . . . . . . . . . . . . . . . . 80 5.2. Resummation via Cayley-Hamilton inverse . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3. Havrda-Charvát-Tsallis entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.4. Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5. Entanglement generated by a time evolution . . . . . . . . . . . . . . . . . . . . . . . . 93 6. Summary and outlook Appendix A. Weingarten calculus for the first steps of the IPR signal function . . . . . . . . . . .99 B. Color-Flavor transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 C. Detailed calculation of moments using SVD . . . . . . . . . . . . . . . . . . . . . . . . 102 D. Integral I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 E. Stationary phase approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 F. Ergodic average of the coupling term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 List of Figures List of Tables / Es wird üblicherweise angenommen, dass die Verschränkungsentropie von Eigenzuständen quantenchaotischer Systeme durch die Theorie der Zufallsmatrizen beschrieben wird. Diese Zufallsmatrixvorhersage bezieht sich nur auf strukturlose Zufallszustände. Es ist nicht klar, wie sich die Subsystemstruktur realer, bipartiter Systeme auf die Verschränkung auswirkt. Wir untersuchen die Konsequenzen einer solchen Struktur auf die bipartite Verschränkung der Eigenzustände von zeit-periodisch gestoßenen Floquet-Systemen. Dazu wird der Ausdruck für die Eigenzustandsverschränkung in eine dynamische Größe überführt, welche besonders geeignet ist für die Anwendung analytischer Methoden zur Zeitentwicklung. Wir präsentieren drei Ansätze und wenden jeden auf ein zugehöriges minimales Modell an. Basierend auf der Supersymmetriemethode berechnen wir die Verschränkung in strukturlosen Zufallsmatrizen und erhalten exakte Resultate für die Entropie von Zufallsmatrixeigenzuständen. Der Weingarten-Formalismus wird genutzt, um die Verschränkung in einem inhärent bipartiten Zufallsmatrixmodell zu berechnen. Außerdem stellen wir, basierend auf semiklassischen Pfad-Integralen, eine Spurformel auf, welche die Verschränkung in chaotischen Floquet-Systemen mittels klassischer Orbits ausdrückt. Wir zeigen über diesen Weg, dass die Verschränkung in stark gekoppelten, bipartiten Floquet-Systemen im semiklassischen Limes mit der Verschränkung in strukturlosen Zufallsmatrizen übereinstimmt. Es werden mehrere Verallgemeinerungen unserer Methoden für autonome Systeme und andere Entropien diskutiert.:1. Introduction 2. Fundamentals on bipartite systems and entanglement 2.1. Classical and quantum chaotic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1. Classical mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2. Quantum systems and random matrix theory . . . . . . . . . . . . . . . . . . . 8 2.2. Bipartite systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3. Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4. Objective of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3. Random matrix methods for entanglement in bipartite chaotic systems 3.1. Entropy formulation in terms of Green’s functions . . . . . . . . . . . . . . . . . . . . . 23 3.2. Weingarten calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1. Spectral form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2. Inverse participation ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.3. Linear entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3. Linear entropy by the supersymmetry method . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.1. Gaussian integrals and the generating function . . . . . . . . . . . . . . . . . . 44 3.3.2. Supersymmetric integrals and generating function . . . . . . . . . . . . . . . . . 46 3.3.3. Entropy of the CUE case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4. Semiclassical method for entanglement in bipartite chaotic systems 4.1. Path integrals and trace formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.1. Path integral formulation of propagators . . . . . . . . . . . . . . . . . . . . . . 60 4.1.2. Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2. Rescaled path integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.1. Spectral form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2.2. Linear entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.3. Order \hbar correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5. Generalizations 5.1. Supersymmetry method for bipartite systems . . . . . . . . . . . . . . . . . . . . . . . 80 5.2. Resummation via Cayley-Hamilton inverse . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3. Havrda-Charvát-Tsallis entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.4. Autonomous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5. Entanglement generated by a time evolution . . . . . . . . . . . . . . . . . . . . . . . . 93 6. Summary and outlook Appendix A. Weingarten calculus for the first steps of the IPR signal function . . . . . . . . . . .99 B. Color-Flavor transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 C. Detailed calculation of moments using SVD . . . . . . . . . . . . . . . . . . . . . . . . 102 D. Integral I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 E. Stationary phase approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 F. Ergodic average of the coupling term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 List of Figures List of Tables