A precise bare simulation approach to the minimization of some distances. Foundations

In information theory -- as well as in the adjacent fields of statistics, machine learning, artificial intelligence, signal processing and pattern recognition -- many flexibilizations of the omnipresent Kullback-Leibler information distance (relative entropy) and of the closely related Shannon entropy have become frequently used tools. To tackle corresponding constrained minimization (respectively maximization) problems by a newly developed dimension-free bare (pure) simulation method, is the main goal of this paper. Almost no assumptions (like convexity) on the set of constraints are needed, within our discrete setup of arbitrary dimension, and our method is precise (i.e., converges in the limit). As a side effect, we also derive an innovative way of constructing new useful distances/divergences. To illustrate the core of our approach, we present numerous examples. The potential for widespread applicability is indicated, too; in particular, we deliver many recent references for uses of the involved distances/divergences and entropies in various different research fields (which may also serve as an interdisciplinary interface)

Topics in Quantum Gravity and Field Theory

This dissertation addresses a variety of open questions in quantum field theory and quantum gravity. The work fits broadly into two categories: attempts to study black holes and brane dynamics in models of quantum gravity, and attempts to study the entangling surface in quantum field theory

Entanglement and Geometry

There is now strong evidence of a deep connection between entanglement in quantum gravity and the geometry of spacetime. In this dissertation, we study multiple facets of this connection. We start by quantifying the entanglement of a scalar quantum field theory as a function of the curvature of its background. We then shift our focus to the AdS/CFT duality, and we prove multiple logical relationships between geometric statements in AdS and entropic statements in the CFT. Many of these proofs work in the presence of quantum corrections, and we prove under which geometric conditions entanglement wedge nesting continues to imply the quantum null energy condition (QNEC) when the CFT is on an arbitrary curved background. We also demonstrate that the non-gravitational limit of the quantum focusing conjecture implies the QNEC, given the same geometric conditions. Next, we prove the connection between the boundary of the future of a surface and the null geodesics originating orthogonally from that surface. This theorem is important for proving that the area of holographic screens increases monotonically. Finally, we derive the holographic prescription for computing Renyi entropies of a CFT with the formalism of quantum error-correction. In the process, we provide evidence that the quantum gravity degrees of freedom related to the AdS geometry are maximally-mixed

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

Holographic Entanglement Entropy: RG Flows and Singular Surfaces

Over the past decade, the AdS/CFT correspondence has proven to be a remarkable tool to study various properties of strongly coupled field theories. In the context of the holography, Ryu and Takayanagi have proposed an elegant method to calculate entanglement entropy for these field theories. In this thesis, we use this holographic entanglement entropy to study a candidate c-theorem and entanglement entropy for singular surfaces. We use holographic entanglement entropy for strip geometry and construct a candidate c-function in arbitrary dimensions. For holographic theories dual to Einstein gravity, this c-function is shown to decrease monotonically along RG flows. A sufficient condition required for this monotonic flow is that the stress tensor of the matter fields driving the holographic RG flow must satisfy the null energy condition over the holographic surface used to calculate the entanglement entropy. In the case where the bulk theory is described by Gauss-Bonnet gravity, the latter condition alone is not sufficient to establish the monotonic flow of the c-function. We also observe that for certain holographic RG flows, the entanglement entropy undergoes a ‘phase transition’ as the size of the system grows and as a result, evolution of the c-function may exhibit a discontinuous drop. Then, we turn towards studying the holographic entanglement entropy for regions with a singular boundary in higher dimensions. Here, we find that various singularities make new universal contributions. When the boundary CFT has an even spacetime dimension, we find that the entanglement entropy of a conical surface contains a term quadratic in the logarithm of the UV cut-off. In four dimensions, the coefficient of this contribution is proportional to the central charge c. A conical singularity in an odd number of spacetime dimensions contributes a term proportional to the logarithm of the UV cut-off. We also study the entanglement entropy for various boundary surfaces with extended singularities. In these cases, extended singularities contribute through new linear or quadratic terms in logarithm only when the locus of the singularity is even dimensional and curved