Divergence Measures

Data science, information theory, probability theory, statistical learning and other related disciplines greatly benefit from non-negative measures of dissimilarity between pairs of probability measures. These are known as divergence measures, and exploring their mathematical foundations and diverse applications is of significant interest. The present Special Issue, entitled “Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems”, includes eight original contributions, and it is focused on the study of the mathematical properties and applications of classical and generalized divergence measures from an information-theoretic perspective. It mainly deals with two key generalizations of the relative entropy: namely, the R_ényi divergence and the important class of f -divergences. It is our hope that the readers will find interest in this Special Issue, which will stimulate further research in the study of the mathematical foundations and applications of divergence measures

Functional inequalities in quantum information theory

Functional inequalities constitute a very powerful toolkit in studying various problems arising in classical information theory, statistics and many-body systems. Extensions of these tools to the noncommutative setting have been introduced in the beginning of the 90's in order to study the asymptotic properties of certain quantum Markovian evolutions. In this thesis, we study various extensions and problems arising from the specific noncommutative nature of such processes. The first logarithmic Sobolev inequality to be proved, due to Gross, was for the Ornstein Uhlenbeck semigroup, that is the Brownian motion with friction on the real line. The generalization of this result to the quantum Ornstein Uhlenbeck semigroup was found very recently by Carlen and Maas, and de Palma and Huber by means of different techniques. The latter proof consists of a quantum generalization of the so-called entropy power inequality. Here, we consider another possible version of the entropy power inequality and use it to derive asymptotic properties of the frictionless quantum Brownian motion. The proof of Carlen and Maas discussed in the previous paragraph relies on their new quantum extension of the classical notion of displacement convexity. This is classically known to imply most of the usual functional inequalities such as the modified logarithmic Sobolev inequality and Poincaré's inequality. Here, we further study the framework introduced by Carlen and Maas. In particular, we show how displacement convexity implies quantum functional and transportation cost inequalities. The latter are then used to derive certain concentration inequalities of quantum states in the spirit of Bobkov and Goetze. These concentration inequalities are used in order to derive finite sample size bounds for the task of quantum parameter estimation. The main advantage of classical logarithmic Sobolev inequalities over other methods resides in their tensorization property: the strong log-Sobolev constant of the product of independent Markovian evolutions is equal to the maximum over the set of strong log-Sobolev constants of the individual evolutions. However, this property is strongly believed to fail in the non-commutative case, due to the non-multiplicativity of noncommutative Lp to Lq norms. In this thesis, we show tensorization of the logarithmic Sobolev constants for the simplest quantum Markov semigroup, namely the generalized depolarizing semigroup. Moreover, we consider a new general method to overcome the issue of tensorization for general primitive quantum Markov semigroups by looking at their contractivity properties under the completely bounded Lp to Lq norms. This method was first investigated in the restricted case of unital semigroups by Beigi and King. Noncommutative functional inequalities considered in the present literature only deal with primitive quantum Markovian semigroups which model memoryless irreversible dynamics converging to a specific faithful state. However, quantum Markov semigroups can in general display a much richer behavior referred to as decoherence: In particular, under some mild conditions, any such semigroup is known to converge to an algebra of observables which effectively evolve unitarily. Here, we introduce the concept of a decoherence-free logarithmic Sobolev inequality, and the related notion of hypercontractivity of the associated evolution, to study the decoherence rate of non-primitive quantum Markov semigroups. Moreover, we utilize the transference method recently introduced by Gao, Junge and LaRacuente, in order to find decoherence times associated to a class of decoherent Markovian evolutions of great importance in the field of quantum error protection, namely collective decoherence semigroups. Finally, we develop the notion of quantum reverse hypercontractivity, first introduced by Cubitt, Kastoryano, Montanaro and Temme in the unital case, and apply it in conjunction with the tensorization of the modified logarithmic Sobolev inequality for the generalized depolarizing semigroup in order to find strong converse rates in quantum hypothesis testing and for the classical capacity of classical-quantum channels. Moreover, the transference method also allows us to find strong converse bounds on the various capacities of quantum Markovian evolutions

Speed of convergence for laws of rare events and escape rates

We obtain error terms on the rate of convergence to Extreme Value Laws for a general class of weakly dependent stochastic processes. The dependence of the error terms on the `time' and `length' scales is very explicit. Specialising to data derived from a class of dynamical systems we find even more detailed error terms, one application of which is to consider escape rates through small holes in these systems

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result

Spectral properties of localized continuum random Schrödinger operators

The results presented in this thesis are mainly motivated by the attempt to improve the mathematical understanding of the localized spectral region of random quantum mechanical systems. It is common wisdom in theoretical (and experimental) physics that a variety of spectral properties are characteristic indicators for the presence of spectral localization. The mathematical verifi of such characteristic properties at large is one of the key concerns of the theory of random Schrödinger operators. The first topic we address, based on joint work with Martin Gebert and Peter Müller [37], is a phenomenon dubbed Anderson orthogonality : Given two non-interacting, quasi-free electron systems which only differ by a local perturbation, Anderson orthogonality refers to the vanishing of their ground-state overlap in the macroscopic limit. We prove that in the localized spectral region Anderson orthogonality and absence of Anderson orthogonality both typically appear with positive probability. As a consequence, the disorder-averaged ground- state overlap does not vanish in the macroscopic limit. Combined with the mathematical results from [51], this shows that the absence of Anderson orthogonality can indeed be viewed as a characteristic property of the localized spectral region. Another test for the spectral structure of a random quantum mechanical system is its local eigenvalue statistics. On the one hand, it is common sense in physics that the eigenvalue statistics for a generic localized system are poissonian. But, on the other hand, previously known proofs only applied for the lattice Anderson model and similar lattice models. Irre- spective of the concrete model, a mandatory requirement to obtain Poisson statistics of the local eigenvalue process around a reference energy E is a positive density of states at that point. As a first step towards Poisson statistics we prove, based on joint work with Martin Gebert, Peter Hislop, Abel Klein and Peter Müller [37], a strictly positive lower bound on the density of states for continuum random Schrödinger operators. Then, based on joint work with Alexander Elgart [36], we present a new proof for poissonian local eigenvalue statistics. It is more flexible than known methods and, for instance, applicable to continuum random Schrödinger operators. A phenomenon reminiscent of the vanishing of the ground-state overlap described above is the logarithmic enhancement of asymptotic Szegő-type trace formulas. The absence of a logarithmic enhancement for the localized lattice Anderson model is already known [100, 43]. But motivated by those works, we prove [35] a full asymptotic expansion for the trace of h(g(Hω )[−L,L]d ) in terms of the length-scale L, where h and g are suitable functions and Hω is a general ergodic operator. Our key assumption here is that the operator kernel of g(Hω ) exhibits sufficient spatial decay, which can be verifi either under a spectral localization assumption on Hω or a regularity assumption on g.Die Resultate, die ich im Rahmen meiner Dissertation vorstelle, sind hauptsächlich motiviert durch das Bestreben, das mathematische Verständnis des lokalisierten Spektralbereichs zufälliger quantenmechanischer Systeme zu verbessern. In der theoretischen (und experimentellen) Physik gelten verschiedene Spektraleigenschaften als charakteristische Indikatoren für das Vorliegen einer lokalisierten spektralen Phase. Das mathematische Bestätigen solcher Charakteristika in möglichst großer Allgemeinheit ist eines der Kernthemen der Theorie zufäl- liger Schrödingeroperatoren. Im ersten Projekt dieser Dissertation, welches auf einer Zusammenarbeit mit Martin Gebert und Peter Müller basiert [37], wird die sogenannte Anderson Orthogonalität unter- sucht: Gegeben seien zwei nicht wechselwirkende Elektronensysteme, deren Einteilchenoperatoren sich nur um eine lokale Störung unterscheiden. Dann spricht man von Anderson Orthogonalität, falls der Überlapp der beiden Grundzustände der Elektronensysteme im makroskop- ischen Limes gegen null strebt. Wir zeigen, dass Anderson Othogonalität sowie deren Ab- wesenheit im lokalisierten Spektralbereich eines zufälligen Schrödingeroperators beide mit positiver Wahrscheinlichkeit auftreten. Folglich verschwindet der zufallsgemittelte Grundzustandsüberlapp nicht im makroskopischen Limes. In Kombination mit bereits bekannten Resultaten [51] zeigt dies, dass das Verhalten des Grundzustandüberlapps im makroskopischen Limes ein Indikator eines lokalisierten Spektralbereichs ist. Ein weiterer Test für die Spektralstruktur eines zufälligen quantenmechanischen Systems ist dessen lokale Eigenwertstatistik. Es ist Teil der Folklore der Physik, dass eine poisson- verteilte lokale Eigenwertstatistik ein universeller Indikator eines lokalisierten Systems ist. Andererseits funktionieren bekannte Beweise nur für das klassiche Andersonmodell und ähnliche Modelle auf dem Gitter. Unabhängig vom jeweiligen Modell ist eine notwendige Bedingung für eine poissonverteilte lokale Eigenwertstatistik bei der Referenzenergie E die strikte Positivität der Zustandsdichte an dieser Energie. Im zweiten Projekt, welches auf einer Zusammenarbeit mit Martin Gebert, Peter Hislop, Abel Klein und Peter Müller basiert [37], wird eine strikt positive untere Schranke an die Zustandsdichte von zufälligen Schrödingeroperatoren im Kontinuum etabliert. Danach präsentiere ich, basierend auf Resultaten die in Zusammenarbeit mit Alexander Elgart entstanden [36], einen neuen Beweis für die poissonsche lokale Eigenwert- statistik. Dieser ist deutlich flexibler als bekannte Beweise und ist zum Beispiel anwendbar auf zufällige Schrödingeroperatoren im Kontinuum. Ein Phänomen, welches dem oben beschriebenen asymptotischen Verschwinden des Grundzustandsüberlapps ähnlich ist, ist die logarithmische Verstärkung der führenden Ord- nung sogenannter asymptotischer Szegő Spurformeln. Die Absenz solcher logarithmischer Verstärkungen für lokalisierte zufällige Schrödingeroperatoren ist bereits bekannt [100, 43]. Auf- bauend auf diesen Arbeiten beweise ich [35] eine komplette asymptotische Entwicklung für die Spur des Operators h(g(Hω )[−L,L]d ) in der Längenskala L, wo h und g geeignete Funktionen sind und Hω ein allgemeiner ergodischer Operator. Die Hauptannahme, unter der diese komplette asymptotische Entwicklung gültig ist, ist hinreichend schneller Abfall des Operatorkerns des Operators g(Hω ). Eine solche Annahme kann nachgewiesen werden unter entweder einer spektralen Lokalisierungsannahme für den Operator Hω oder einer Regularitätsannahme für die Funktion g