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Perhaps the deepest unanswered question in theoretical physics is how to consistently combine the theory of general relativity with the principles of quantum mechanics. In parallel, the holographic principle is expected to be a fundamental property of quantum gravity that allows the latter to be described by a theory in one dimension lower which does not contain gravity. As an explicit realization of the holographic principle, the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence has established itself as our best non-perturbative approach to quantum gravity. Moreover, it has become clear that concepts of quantum information theory play a fundamental role in this duality. Nonetheless, the extent of validity of the holographic principle has not been fully explored yet. In parallel, the AdS/CFT correspondence is still primarily a conjecture rooted in theoretical arguments and, while awaiting a mathematical proof, it would immensely benefit from experimental tests validating its predictions. Motivated by these aspects, in this thesis I advance the field of \textit{discrete holography} by establishing explicit holographic models on both bulk and boundary of discretizations of hyperbolic space. These models assess the wide range of validity of the holographic principle beyond the continuum formulation of AdS/CFT while also enabling the experimental realization of holographic predictions. Furthermore, I investigate quantum information quantities such as circuit complexity, quantum discord, and the operator algebra description of entanglement, and elucidate their roles in describing certain properties of black holes. In the first part of this thesis I consider regular tilings of hyperbolic space as a discretization of constant-time slices of AdS$_{3}$. From the perspective of the bulk, I show that they provide a sensible description of the continuum by establishing how the Breitenlohner-Freedman stability bound for a free scalar field is realized on these tilings. I follow up on this result by considering an interacting scalar field on hyperbolic tilings and showing how a discrete lattice model can emulate CFT correlation functions at both zero and finite temperature. The latter are obtained from a discretization of a constant-time slice of an AdS black hole. Additionally, I present a universal experimental platform based on electric circuits on which the aforementioned results can be measured. My results show that these architectures bear tremendous untapped potential to realize further holographic predictions in the laboratory, beyond those appraised in this thesis. From a boundary perspective, I establish a large class of explicit Hamiltonians defined at the asymptotic boundary of hyperbolic tilings. These are prime candidates for boundary theories in discrete holographic dualities and consist of aperiodically disordered quantum spin chains with nearest-neighbor interactions. The aperiodic disorder encodes information about the systematic construction of the bulk tiling based on substitution rules. Using real-space renormalization group techniques, I study the ground state of these models in view of their factorized form into two-spin states, their correlation functions, and their entanglement structure. I show how the latter is exactly captured by a tensor network which extends into the bulk, thus providing a geometric bulk dual to the boundary theory. The high degree of analytic tractability of these models enables discrete manifestations of the holographic principle. These are realized by the standard prescription for computing entanglement entropy in tensor networks. In the second part of this thesis I focus on three quantum information-theoretic quantities and study them in view of describing properties of quantum black holes via the AdS/CFT correspondence. Starting from quantum circuit complexity as a conjectured probe for the emergence of space behind the horizon, I study a novel definition of Nielsen operator complexity for finite- and infinite-dimensional systems. My construction relates complexity with the theory of classical ideal hydrodynamics while still exhibiting the features desired for a holographic complexity measure. Then, I study geometric quantum discord as an efficient tool for diagnosing non-factorization. This quantity is easier to calculate than its NP-hard cousin (quantum discord), and I show how the geometric quantum discord for pure states vanishes if and only if the modular partition function factorizes. I apply this to the explicit example of an eternal wormhole in AdS and discuss the results in view of holography. Moreover, I establish a relation between geometric quantum discord and wormhole contributions to the gravitational path integral, thus shedding light on the mechanisms behind the factorization puzzle of AdS/CFT. Finally, I study local operator algebras as a rigorous approach to entanglement entropy which has been recently employed in AdS/CFT to explain the emergence of time behind the black hole horizon. In particular, I consider operator algebras in an interacting but exactly solvable quantum many-body system of Majorana fermions. I find transitions between different types of operator algebras throughout the phase diagram of the model, which contains a quantum phase transition. This model provides an explicit instance where such operator algebra transitions can be studied in a tractable many-body system. The results presented in this thesis are published in my works listed in App.~A. / Die wahrscheinlich grundlegendste unbeantwortete Frage der theoretischen Physik betrifft die Vereinigung der allgemeinen Relativitätstheorie mit den Prinzipien der Quantenmechanik. Einer der vielversprechendsten Zugänge beruht auf dem holographischen Prinzip, das eine fundamentale Eigenschaft der Quantengravitation zu sein scheint. Es besagt, dass sich eine Gravitationstheorie alternativ durch eine Theorie ohne Gravitation in einer Dimension niedriger beschreiben lässt. Als explizite Realisierung des holographischen Prinzips hat sich die Anti-de Sitter/Konforme Feldtheorie (AdS/CFT) Korrespondenz als führender Ansatz zur Theorie der Quantengravitation etabliert. Darüber hinaus haben sich Konzepte aus der Quanteninformationstheorie als elementare Bestandteile dieser Dualität heraus\-kristallisiert. Dennoch ist der Geltungsbereich des holographischen Prinzips bisher nicht vollständig erforscht. Zudem ist die AdS/CFT Korrespondenz weitestgehend noch eine Vermutung. Solange ein mathematischer Beweis nicht zur Verfügung steht, kann sie von experimentellen Tests ihrer Vorhersagen extrem profitieren. Diese Aspekte motivieren die aktuelle Dissertation, in der ich Fortschritte im Forschungsfeld der \textit{diskreten Holographie} präsentiere. Dazu etabliere ich explizite holographische Modelle auf Diskretisierungen des hyperbolischen Raums. Diese Modelle zeigen einen erweiterten Gültigkeitsbereich des holographischen Prinzips auf, jenseits der kontinuierlichen Formulierung von AdS/CFT. Zudem ermöglichen sie die experimentelle Realisierung holographischer Vorhersagen. Weiterhin präsentiere ich meine Ergebnisse zu Quanteninformationsmaßen wie der Komplexität von Quantenschaltkreisen, dem Quantendiskord und der Beschreibung von Verschränkungsentropie mittels Operatoralgebren. Ebenso erörtere ich die Signifikanz dieser Maße für die holographische Beschreibung ausgewählter Eigenschaften von Schwarzen Löchern. Im ersten Teil dieser Dissertation beschäftige ich mich mit regulären Tessellierungen des hyperbolischen Raums als Diskretisierungen von AdS$_3$-Hyperflächen bei konstanter Zeit. Für den AdS Raum zeige ich, dass diese Tessellierungen das Kontinuum sinnvoll nähern, indem ich die Breitenlohner-Freedman-Stabilitätsgrenze eines freien Skalarfeldes auf solchen Diskretisierungen realisiere. Anknüpfend an dieses Ergebnis betrachte ich ein wechselwirkendes Skalarfeld auf hyperbolischen Tessellierungen und zeige, wie ein diskretes Gittermodell Korrelationsfunktionen der CFT sowohl bei verschwindender als auch bei endlicher Temperatur emulieren kann. Die Korrelationsfunktionen bei endlicher Temperatur erhalte ich aus der Diskretisierung einer Hyperfläche eines Schwarzen Lochs im AdS-Raum bei konstanter Zeit. Darüber hinaus stelle ich eine universelle experimentelle Plattform vor, die auf elektrischen Schaltkreisen basiert und auf der die oben genannten Ergebnisse gemessen werden können. Meine Ergebnisse zeigen, dass diese Architekturen ein enormes ungenutztes Potenzial aufweisen, um weitere holographische Vorhersagen im Labor zu realisieren, welche über die in dieser Arbeit getesteten hinausgehen. Für die Quantentheorie am Rand des AdS-Raums stelle ich eine große Klasse expliziter Hamilton-Operatoren auf, die am asymptotischen Rand hyperbolischer Tessellierungen definiert sind. Diese sind geeignete Kandidaten für Randtheorien in diskreten \mbox{holographischen} Dualitäten und bestehen aus aperiodisch ungeordneten Quantenspinketten mit Wechselwirkungen zwischen nächsten Nachbarn. Die aperiodische Unordnung speichert Informationen über den systematischen Aufbau der Tessellierung. Mit Hilfe von Realraum-Renormierungsgruppen-Techniken untersuche ich den Grund\-zustand dieser Modelle im Hinblick auf ihre faktorisierte Form in Zwei-Spin-Zustände, ihre Korrelations\-funktionen und ihre Verschränkungsstruktur. Ich zeige, dass diese Struktur exakt durch ein Tensornetzwerk erfasst wird, das sich zurück in den AdS-Raum erstreckt und somit ein geometrisches Dual zur Randtheorie liefert. Die analytische Nachvoll\-ziehbarkeit dieser Modelle ermöglicht es aufzuzeigen, wie diese das holographische Prinzip für diskrete Systeme realisieren. Dies erziele ich durch die Standardvorschrift zur Berechnung der Verschränkungsentropie in Tensornetzwerken. Im zweiten Teil dieser Arbeit konzentriere ich mich auf drei quanteninformationstheoretische Größen und untersuche sie im Hinblick auf die Beschreibung von Schwarzen Löchern mittels der AdS/CFT-Korrespondenz. Ausgehend von der Komplexität von Quantenschaltkreisen, die als Maß für den Raum hinter dem Horizont vorgeschlagen wird, untersuche ich eine neuartige Definition der Nielsen-Operator-Komplexität für endlich- und unendlich-dimensionale Systeme. Mein Ansatz verbindet die Komplexität mit der Theorie der klassischen idealen Hydrodynamik und weist gleichzeitig die Eigenschaften auf, die man sich von einem holographischen Komplexitätsmaß wünscht. Anschließend untersuche ich den geometrischen Quantendiskord als effizientes Werkzeug zur Diagnose von Nicht-Faktorisierung. Diese Größe ist einfacher zu berechnen als ihr NP-schwerer Verwandter, der (nicht-geometrische) Quantendiskord. Darüber hinaus zeige ich, dass der geometrische Quantendiskord für reine Zustände genau dann verschwindet, wenn die modulare Zustandssumme faktorisiert. Ich wende dies auf das explizite Beispiel eines zeitunabhängigen Wurmlochs im AdS-Raum an und diskutiere die holographischen Eigenschaften der Ergebnisse. Daraufhin stelle ich eine Beziehung zwischen geometrischem Quantendiskord und Wurmloch-Beiträgen zum Gravitationspfadintegral her und beleuchte damit die möglichen Mechanismen hinter dem AdS/CFT-Faktorisierungsproblem. Schließlich untersuche ich lokale Operatoralgebren als mathematischen Ansatz für die Verschränkungsentropie. Diese wurden kürzlich in AdS/CFT verwendet, um die Zeitentwicklung im Inneren des Schwarzen Lochs zu erklären. Insbesondere betrachte ich Operatoralgebren in einem wechselwirkenden, aber dennoch exakt lösbaren Quanten-Vielkörpersystem von Majorana-Fermionen. Im gesamten Phasendiagramm des Modells, das einen Quantenphasenübergang enthält, finde ich Übergänge zwischen verschiedenen Typen von Operatoralgebren. Somit stelle ich ein explizites und nachvollziehbares Vielteilchenmodell dar, in dem solche Algebra-Übergänge realisiert werden. Die in dieser Dissertation präsentierten Ergebnisse sind in meinen in Anhang~A aufgelisteten Originalveröffentlichungen publiziert.

Este é um trabalho de revisão. Os principais formalismos no desenvolvimento da teoria de cordas são introduzidos e discutidos: a corda bosônica e as supercordas no formalismo de Ramond-Neveu-Schwarz e no formalismo de Green-Schwarz. São introduzidas também as açôes efetivas no limite de baixas energias das D-branas (ação DBI) e das teorias de supergravidade (D=10,11). Por fim, são analisados alguns aspectos da correspondência AdS/CFT, enunciando a conjectura e introduzindo algumas das entradas do dicionário. Os apêndices contêm tópicos de grande relevância para o texto e podem ajudar a esclarecer vários raciocínios e passagens ao longo do mesmo. / This is a review work. The main formalisms on the development of string theory are introduced and discussed: the bosonic string and the superstrings formalisms of Ramond-Never-Schwarz and Green-Schwarz. The effective actions in the low energy limit of the D-branes (DBI action) and supergravity theories (D=10,11) are introduced as well. Finally, some aspects of the AdS/CFT correspondence are analised, enunciating the conjecture and introducing some entries of the dictionary. The appendices contains some topics that are highly relevant for the whole text and may help clarify several ideas and arguments through it.

Esta tese é um estudo de estabilidade de modos quasinormais em um sistema que apresenta uma mudança de estabilidade ao variar continuamente os parâmetros físicos de um buraco negro. A mudança de estabilidade encontrada possui interpretações na correspondência AdS/CFT. A ferramenta principal utilizada neste trabalho para o cálculo de modos quasinormais foram métodos numéricos que podem ser utilizados em inúmeros trabalhos desta área de pesquisa, especialmente por não dependerem de suposições de simetria ou de comportamento conveniente do sistema físico. / This thesis is a study of stability of quasinormal modes in a system featuring a stability change if one continuously varies the physical parameters of a black hole. The stability change thus found has some possible interpretations in the AdS/CFT correspondence. The main tool used in this study for calculating quasinormal modes are numerical methods that can be used in numerous works in this area of research, especially because they do not depend on assumptions of symmetry or any convenient behavior of the physical system.

O escopo deste mestrado é de se familiarizar com a chamada {\\it correspondência AdS/CFT}, que tem sido um dos mais importantes desenvolvimentos na física teórica nas últimas décadas. De acordo com essa correspondência, deformações das geometrias do lado da gravidade (ou lado \"AdS\") devem ser mapeadas para operadores das teorias de calibre duais (ou lado \"CFT\"). Em particular, nos temos estado interessados em explorar uma entrada particular no dicionário AdS/CFT, a relação entre os operadores 1/2 BPS em ${\\cal N}=4$ super Yang-Mills, e as chamadas {\\it geometrias bubbling} no lado da gravidade. A fim de fazer isso, apresentamos primeiramente as noções de ${\\cal N}=4$ SYM e soluções de Supergravidade. Portanto, podemos expor mais claramente o sentido da correspondência AdS /CFT, e depois mostrar a derivação das geometrias 1/2 BPS duais a estados 1/2 BPS em ${\\cal N}=4$ SYM como um exemplo. / The scope of this Master program was to get acquainted with the so-called {\\it AdS/CFT correspondence}, which has been one of the most important developments in theoretical physics in the last decades. According to this correspondence, deformations of the geometries in the gravity side (or \"AdS\" side) must be mapped to states of the dual gauge theories (or \"CFT\" side). In particular, we have been interested in exploring a particular entry in the AdS/CFT dictionary, namely, the relation between 1/2 BPS operators in ${\\cal N}=4$ super Yang-Mills, and the so-called {\\it bubbling geometries} on the gravity side. In order to do that, we first present the notions of N=4 SYM and Supergravity solutions. In this way, we can expose the statement of the AdS/CFT correspondence, and later show the derivation of 1/2 BPS geometries dual to 1/2 BPS states in N=4 SYM as an example of this one.

A written report of a paper titled Holographic dual of collimated radiation by Veronika E. Hubeny where a new and easier method is proposed to estimate the “radiation due to an accelerated quark in a strongly coupled medium”. The method is able to reproduce the results from an earlier paper without the need of solving the linearized Einstein equations but by way of calculating geodesics in AdS using the AdS/CFT correspondence and the gravitational dual of the quark being a string. A quick introduction to synchrotron radiation and general relativity is given after which the AdS/CFT correspondence is introduced along with the results and method of V. Hubeny. / A bachelor thesis in theoretical physics.

Este é um trabalho de revisão. Os principais formalismos no desenvolvimento da teoria de cordas são introduzidos e discutidos: a corda bosônica e as supercordas no formalismo de Ramond-Neveu-Schwarz e no formalismo de Green-Schwarz. São introduzidas também as açôes efetivas no limite de baixas energias das D-branas (ação DBI) e das teorias de supergravidade (D=10,11). Por fim, são analisados alguns aspectos da correspondência AdS/CFT, enunciando a conjectura e introduzindo algumas das entradas do dicionário. Os apêndices contêm tópicos de grande relevância para o texto e podem ajudar a esclarecer vários raciocínios e passagens ao longo do mesmo. / This is a review work. The main formalisms on the development of string theory are introduced and discussed: the bosonic string and the superstrings formalisms of Ramond-Never-Schwarz and Green-Schwarz. The effective actions in the low energy limit of the D-branes (DBI action) and supergravity theories (D=10,11) are introduced as well. Finally, some aspects of the AdS/CFT correspondence are analised, enunciating the conjecture and introducing some entries of the dictionary. The appendices contains some topics that are highly relevant for the whole text and may help clarify several ideas and arguments through it.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, estudamos numericamente a formaÃÃo e a dinÃmica de defeitos topolÃgicos do tipo parede de domÃnio em superfluidos em um modelo (3+1)-dimensional do tipo Maxwell-Higgs abeliano, no contexto da correspondÃncia AdS/CFT. Obtemos os propagadores bulk-fronteira, assim como as funÃÃes de correlaÃÃo na fronteira, para um campo escalar nos casos massivo e nÃo-massivo. Se impusermos que os campos dependem apenas da coordenada radial no bulk e de uma coordenada na fronteira, conseguimos obter na teoria dual as soluÃÃes do tipo parede de domÃnio encontradas na literatura. A partir dessas soluÃÃes, estudamos suas propriedades, verificando que o superfluido à caracterizado por duas escalas de comprimento, uma para o parÃmetro de ordem e outra para a densidade de carga. Estudamos tambÃm a variaÃÃo da densidade de carga na regiÃo prÃxima à interface do defeito, e por fim, investigamos as modificaÃÃes necessÃrias na teoria para se estudar a dinÃmica dessas soluÃÃes. / We numerically study the formation and dynamics of domain-wall-like topological defects in superfluids, using a (3+1)-dimensional abelian Maxwell-Higgs model, under the AdS/CFT correspondence. We obtain the bulk-boundary propagators, as well as the correlation functions on the boundary for a scalar field, both in massive and non-massive cases. If we impose that the fields depends only on a boundary coordinate and the bulk radial coordinate, we obtain in the dual theory the domain wall solutions found in literature. From these solutions we found that the superfluid is characterized by two length scales, one for the order parameter and other for the charge density. We also study the change of charge density in the region near the interface of the defect, and finally we investigate the modifications needed in theory to study the dynamics of these solutions.

Esta tese é um estudo de estabilidade de modos quasinormais em um sistema que apresenta uma mudança de estabilidade ao variar continuamente os parâmetros físicos de um buraco negro. A mudança de estabilidade encontrada possui interpretações na correspondência AdS/CFT. A ferramenta principal utilizada neste trabalho para o cálculo de modos quasinormais foram métodos numéricos que podem ser utilizados em inúmeros trabalhos desta área de pesquisa, especialmente por não dependerem de suposições de simetria ou de comportamento conveniente do sistema físico. / This thesis is a study of stability of quasinormal modes in a system featuring a stability change if one continuously varies the physical parameters of a black hole. The stability change thus found has some possible interpretations in the AdS/CFT correspondence. The main tool used in this study for calculating quasinormal modes are numerical methods that can be used in numerous works in this area of research, especially because they do not depend on assumptions of symmetry or any convenient behavior of the physical system.

O escopo deste mestrado é de se familiarizar com a chamada {\\it correspondência AdS/CFT}, que tem sido um dos mais importantes desenvolvimentos na física teórica nas últimas décadas. De acordo com essa correspondência, deformações das geometrias do lado da gravidade (ou lado \"AdS\") devem ser mapeadas para operadores das teorias de calibre duais (ou lado \"CFT\"). Em particular, nos temos estado interessados em explorar uma entrada particular no dicionário AdS/CFT, a relação entre os operadores 1/2 BPS em ${\\cal N}=4$ super Yang-Mills, e as chamadas {\\it geometrias bubbling} no lado da gravidade. A fim de fazer isso, apresentamos primeiramente as noções de ${\\cal N}=4$ SYM e soluções de Supergravidade. Portanto, podemos expor mais claramente o sentido da correspondência AdS /CFT, e depois mostrar a derivação das geometrias 1/2 BPS duais a estados 1/2 BPS em ${\\cal N}=4$ SYM como um exemplo. / The scope of this Master program was to get acquainted with the so-called {\\it AdS/CFT correspondence}, which has been one of the most important developments in theoretical physics in the last decades. According to this correspondence, deformations of the geometries in the gravity side (or \"AdS\" side) must be mapped to states of the dual gauge theories (or \"CFT\" side). In particular, we have been interested in exploring a particular entry in the AdS/CFT dictionary, namely, the relation between 1/2 BPS operators in ${\\cal N}=4$ super Yang-Mills, and the so-called {\\it bubbling geometries} on the gravity side. In order to do that, we first present the notions of N=4 SYM and Supergravity solutions. In this way, we can expose the statement of the AdS/CFT correspondence, and later show the derivation of 1/2 BPS geometries dual to 1/2 BPS states in N=4 SYM as an example of this one.