@phdthesis{Northe2019,
  author    = {Northe, Christian},
  title     = {Interfaces and Information in Gauge/Gravity Duality},
  doi       = {10.25972/OPUS-19159},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-191594},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2019},
  abstract  = {This dissertation employs gauge/gravity duality to investigate features of ( 2 + 1 ) -dimensional quantum gravity in Anti-de Sitter space (AdS) and its relation to conformal field theory (CFT) in 1 + 1 dimensions. Concretely, we contribute to research on the frontier of gauge/gravity with condensed matter as well as the frontier with quantum informa- tion. The first research topic of this thesis is motivated by the Kondo model, which describes the screening of magnetic impurities in metals by conduction electrons at low temperatures. This process has a de- scription in the language of string theory via fluctuating surfaces in spacetime, called branes. At high temperatures the unscreened Kondo impurity is modelled by a stack of pointlike branes. At low tempera- tures this stack condenses into a single spherical, two-dimensional brane which embodies the screened impurity. This thesis demonstrates how this condensation process is naturally reinvoked in the holographic D1/D5 system. We find brane configu- rations mimicking the Kondo impurities at high and low energies and establish the corresponding brane condensation, where the brane grows two additional dimensions. We construct supergravity solutions, which fully take into account the effect of the brane on its surrounding space- time before and after the condensation takes place. This enables us to compute the full impurity entropies through which we confirm the validity of the g-theorem. The second research topic is rooted in the connection of geometry with quantum information. The motivation stems from the "complexity equals volume" proposal, which relates the volume of wormholes to the cicruit complexity of a thermal quantum state. We approach this proposal from a pragmatic point of view by studying the properties of certain volumes in gravity and their description in the CFT. We study subregion complexities, which are the volumes of the re- gions subtended by Ryu-Takayanagi (RT) geodesics. On the gravity side we reveal their topological properties in the vacuum and in ther- mal states, where they turn out to be temperature independent. On the field theory side we develop and proof a formula using kinematic space which computes subregion complexities without referencing the bulk. We apply our formula to global AdS 3 , the conical defect and a black hole. While entanglement, i.e. minimal boundary anchored geodesics, suffices to produce vacuum geometries, for the conical defect we also need geodesics windings non-trivially around the singularity. The black hole geometry requires additional thermal contributions.},
  subject      = {Information},
  language  = {en}
}
@phdthesis{Abt2019,
  author    = {Abt, Raimond},
  title     = {Implementing Aspects of Quantum Information into the AdS/CFT Correspondence},
  doi       = {10.25972/OPUS-18801},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-188012},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2019},
  abstract  = {In recent years many discoveries have been made that reveal a close relation between quantum information and geometry in the context of the AdS/CFT correspondence. In this duality between a conformal quantum field theory (CFT) and a theory of gravity on Anti-de Sitter spaces (AdS) quantum information quantities in CFT are associated with geometric objects in AdS. Subject of this thesis is the examination of this intriguing property of AdS/CFT. We study two central elements of quantum information: subregion complexity -- which is a measure for the effort required to construct a given reduced state -- and the modular Hamiltonian -- which is given by the logarithm of a considered reduced state. While a clear definition for subregion complexity in terms of unitary gates exists for discrete systems, a rigorous formulation for quantum field theories is not known. In AdS/CFT, subregion complexity is proposed to be related to certain codimension one regions on the AdS side. The main focus of this thesis lies on the examination of such candidates for gravitational duals of subregion complexity. We introduce the concept of \textit{topological complexity}, which considers subregion complexity to be given by the integral over the Ricci scalar of codimension one regions in AdS. The Gauss-Bonnet theorem provides very general expressions for the topological complexity of CFT\(_2\) states dual to global AdS\(_3\), BTZ black holes and conical defects. In particular, our calculations show that the topology of the considered codimension one bulk region plays an essential role for topological complexity. Moreover, we study holographic subregion complexity (HSRC), which associates the volume of a particular codimension one bulk region with subregion complexity. We derive an explicit field theory expression for the HSRC of vacuum states. The formulation of HSRC in terms of field theory quantities may allow to investigate whether this bulk object indeed provides a concept of subregion complexity on the CFT side. In particular, if this turns out to be the case, our expression for HSRC may be seen as a field theory definition of subregion complexity. We extend our expression to states dual to BTZ black holes and conical defects. A further focus of this thesis is the modular Hamiltonian of a family of states \(\rho_\lambda\) depending on a continuous parameter \(\lambda\). Here \(\lambda\) may be associated with the energy density or the temperature, for instance. The importance of the modular Hamiltonian for quantum information is due to its contribution to relative entropy -- one of the very few objects in quantum information with a rigorous definition for quantum field theories. The first order contribution in \(\tilde{\lambda}=\lambda-\lambda_0\) of the modular Hamiltonian to the relative entropy between \(\rho_\lambda\) and a reference state \(\rho_{\lambda_0}\) is provided by the first law of entanglement. We study under which circumstances higher order contributions in \(\tilde{\lambda}\) are to be expected. We show that for states reduced to two entangling regions \(A\), \(B\) the modular Hamiltonian of at least one of these regions is expected to provide higher order contributions in \(\tilde{\lambda}\) to the relative entropy if \(A\) and \(B\) saturate the Araki-Lieb inequality. The statement of the Araki-Lieb inequality is that the difference between the entanglement entropies of \(A\) and \(B\) is always smaller or equal to the entanglement entropy of the union of \(A\) and \(B\). Regions for which this inequality is saturated are referred to as entanglement plateaux. In AdS/CFT the relation between geometry and quantum information provides many examples for entanglement plateaux. We apply our result to several of them, including large intervals for states dual to BTZ black holes and annuli for states dual to black brane geometries.},
  subject      = {AdS-CFT-Korrespondenz},
  language  = {en}
}