The work presented in this thesis is a contribution towards the understanding of holo- graphic entanglement from an operational perspective. Following the interpretation which is most natural for quantum information theory, entanglement is viewed as a resource that can be produced, stored, transferred and used for practical purposes. The first chapter introduces the main concepts which are necessary throughout the discussion. Quantum entanglement, as opposed to merely classical correlation, is presented in detail within the framework of quantum mechanics. This is followed by a brief overview on the current state of knowledge about entanglement in quantum field theory and more specifically in gauge/gravity duality. The second chapter investigates a particular measure of entanglement, known as nega- tivity, in the context of holographic field theories. This is further explored in the following (third) chapter, where an interesting dependence of the entanglement between a region and its complement on the topology of their interface is presented. The forth chapter investigates qubits systems and compares equivalence classes of entanglement structures to known properties of holographic states. The fifth chapter focuses on the behaviour of the tripartite information for highly en- tangled states, both in a bipartite and multipartite sense, in relation to the sign definiteness imposed by holography. A final chapter comments on future directions of investigation within this program.
The work presented in this thesis is a contribution towards the understanding of holo- graphic entanglement from an operational perspective. Following the interpretation which is most natural for quantum information theory, entanglement is viewed as a resource that can be produced, stored, transferred and used for practical purposes. The first chapter introduces the main concepts which are necessary throughout the discussion. Quantum entanglement, as opposed to merely classical correlation, is presented in detail within the framework of quantum mechanics. This is followed by a brief overview on the current state of knowledge about entanglement in quantum field theory and more specifically in gauge/gravity duality. The second chapter investigates a particular measure of entanglement, known as nega- tivity, in the context of holographic field theories. This is further explored in the following (third) chapter, where an interesting dependence of the entanglement between a region and its complement on the topology of their interface is presented. The forth chapter investigates qubits systems and compares equivalence classes of entanglement structures to known properties of holographic states. The fifth chapter focuses on the behaviour of the tripartite information for highly en- tangled states, both in a bipartite and multipartite sense, in relation to the sign definiteness imposed by holography. A final chapter comments on future directions of investigation within this program.
This thesis consists of four research papers and an introduction covering the most important concepts appearing in the papers. The papers deal with applications of gauge/gravity dualities in the study of various physical quantities and systems. Gauge/gravity dualities are equivalences between certain quantum field theories and classical theories of gravity. These dualities can be used as computational tools in a wide range of applications across different fields of physics, and as such they have garnered much attention in the last two decades. The great promise of these new tools is the ability to tackle difficult problems in strongly interacting quantum field theories by translating them to problems in classical gravity, where progress is much easier to make. Quantum information theory studies the information contained in quantum systems. Entanglement is the fundamental property of quantum mechanics that sets it apart from classical theories of physics. Entanglement is commonly quantified by entanglement entropy, a quantity which is difficult to compute in interacting quantum field theories. Gauge/gravity dualities provide a practical way for computing the entanglement entropy via the Ryu-Takayanagi formula. The primary focus of this thesis is to use this formula for computing various entanglement measures in strongly interacting quantum field theories via their gravity duals. The purpose of this thesis is to introduce quantum information theory concepts that have been important in our research. When applicable, quantities of interest are first introduced in the classical setting in order to build intuition about their behaviour. Quantum properties of entanglement measures are discussed in detail, along with their holographic counterparts, and remarks are made concerning their applications.
Entanglement entropy is a proposal to quantify quantum entanglement of two disjoint regions in a pure system. It is a relatively new topic and is developing rapidly. The current main motives to study it are its applications to studying quantum gravity, thermalization and phase transitions in condensed matter systems. Mutual information is a related quantity and it can be used to measure the amount of information two disjoint regions share. The purpose of this thesis is to give an introduction to the general results of the field without considering specific systems. The two prominent approaches used are two-dimensional conformal field theory and the holographic entanglement entropy conjecture. The first approach is to calculate entanglement entropy using conformal field theory, the results of which are known to be exact although technically more difficult to derive and only available for 1+1 dimensional systems. Both static and dynamic systems will be discussed. The results are reproduced and generalized to higher dimensions using holography. As a more recent topic, the holographic approach is used to rederive the entanglement entropy of systems with a Fermi surface using AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation. The final chapter of the thesis discusses mutual information in various static and dynamic cases considered in the previous chapters. For the first time, the evolution of mutual information is calculated in AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation in the critical theta=d-1 case.
In the long-standing quest to reconcile gravity with quantum mechanics, profound connections have been unveiled between concepts traditionally pertaining to a quantum information theory, such as entanglement, and constitutive features of gravity, like holography. Developing and promoting these connections from the conceptual to the operational level unlock access to a powerful set of tools which can be pivotal toward the formulation of a consistent theory of quantum gravity. Here, we review recent progress on the role and applications of quantum informational methods, in particular tensor networks, for quantum gravity models. We focus on spin network states dual to finite regions of space, represented as entanglement graphs in the group field theory approach to quantum gravity, and illustrate how techniques from random tensor networks can be exploited to investigate their holographic properties. In particular, spin network states can be interpreted as maps from bulk to boundary, whose holographic behavior increases with the inhomogeneity of their geometric data (up to becoming proper quantum channels). The entanglement entropy of boundary states, which are obtained by feeding such maps with suitable bulk states, is then proved to follow a bulk area law with corrections due to the entanglement of the bulk state. We further review how exceeding a certain threshold of bulk entanglement leads to the emergence of a black hole-like region, revealing intriguing perspectives for quantum cosmology.
We explore aspects of perturbative quantum field theory towards the goal of illuminating features of quantum gravity. In Chapter 1, we begin with the color-kinematic duality, a surprising relationship between gauge theoretic and kinematic properties of scattering amplitudes that, via the double-copy property, leads to a deep connection between gauge theory and gravity. We investigate the relationship between the color-kinematic duality and amplitude relations in ABJM theory and find that the duality is satisfied eight points without associated amplitude relations present in all other known instances of the duality. In Chapter 2, we derive a basic consistency check for AdS3/CFT2, a tractable example of the well-known AdS/CFT duality. We show that, under certain mild assumptions on the light spectrum of the CFT, CFT correlators of light operators match those computed in AdS perturbatively in 1/N: in a black hole background for high temperatures, and in thermal AdS for low temperatures. Next we turn to entanglement entropy, an information-theoretic quantity that in the CFT may encode dual AdS geometry. Time-dependent entanglement entropy has been studied for generic excited states. However, localized unitary operators in particular are in correspondence with Hamiltonian perturbations, and are basic building blocks of local excitations. In Chapter 3, we detail these operators' locality properties as well as the differences between excited states created by Hermitian operators and those created with localized unitary operators. In Chapter 4, we initiate the perturbative exploration of entanglement entropy with a time-dependent Hamiltonian, computing past first order in perturbation theory. We find a universal structure of entanglement propagation: interactions entangle unentangled excitations according to entanglement diagrams that track the flow of entanglement. We provide diagrammatic tools to simplify computations of loop entanglement diagrams.
Holographic duality describes gravitational theories in terms of quantum many-body systems. In holography, quantum information theory provides a crucial tool that directly connects microscopic structures of these systems to the geometries of gravitational spacetimes. One manifestation is that the entanglement entropy in quantum many-body systems can be calculated from the area of an extremal surface in the corresponding gravitational spacetime. This implies that a gravitational spacetime can emerge from an enormous number of entangled qubits. In this Essay, I will discuss open problems in this area of research, considering recent developments and outlining future prospects towards a complete understanding of quantum gravity. The first step in this direction is to understand what kind of quantum circuits each holographic spacetime corresponds to, drawing on recent developments in quantum complexity theories and studying concrete examples of holography in string theory. Next, we should extend the concept of holography to general spacetimes, e.g., those spacetimes which appear in realistic cosmologies, by utilizing the connections between quantum information and holography. To address the fundamental question of how time emerges, I will propose the concepts of pseudoentropy and timelike entanglement as a useful tool in our exploration. Part of a series of Essays in Physical Review Letters which concisely present author visions for the future of their field.
In this thesis we will explore entanglement of various subsystems in quantum field theory, and its uses for describing underlying structure of states. We begin this journey by focusing on topological quantum field theory in chapters 1 and 2. These chapters are based heavily upon the papers [1] and [2, 3], respectively. In the introduction to chapter 1 we take the opportunity to recall the definition of bipartite entanglement entropy in quantum field theory and describe, briefly, how the entropy associated to connected spatial subregions provides a signature of topological order at the level of the wave-function. In particular, for a gapped phase with topological order, we recapitulate how the sub-leading correction to the “area-law” of the entanglement entropy contains coarse information about the topological phase while also being insensitive to the short-distance details of the underlying theory. We then present results that show that this sub-leading correction can also detect the class of interactions that glue a subregion to its complement. In principle, the subregion could be a different topological phase than its complement, and so we say that the entanglement entropy provides a signature of the interface between topological phases. In showing this, we discuss the classes of interactions between gapped topological phases and how they are represented as boundary conditions in the low-energy effective topological field theory, which in this case is an Abelian Chern-Simons theory. Using these boundary conditions, we perform path-integral calculations of the entanglement entropy using the replica trick and show that the universal contribution does indeed depend on the class of boundary conditions, and the gapped topological phases of the subsystem and its complement. To finish this chapter we reformulate this problem at the level of the Hilbert space and show how these universal contributions are related to resolving a generic ambiguity in defining subsystems of field theories with gauge invariance. We continue in the theme of topological field theories for chapter 2. Here we will shift focus from subregions of a connected spatial slice to disconnected spatial regions in an effort to explore, firstly, how long-range entanglement is encoded in topological field theory, and secondly, how that entanglement is encoded amongst multiple parties. We begin with a brief introduction of multi-party entanglement and recall the classification of three-party entanglement. Moving on to our specific setup, we describe the construction of states on multiple non-intersecting torus boundaries. Generically, these are states on link complements where the torus boundaries can be thought of as the edges of thickened closed strings entwined and knotted inside a compact manifold, which we will take to be the three-sphere. The wave-functions of these states are represented by various link invariants (such as the Gauss linking number for Abelian links, and Jones polynomial for SU(2)). We will exhibit several examples of multi-component links in both the Abelian and the SU(2) theories, and afterwards discuss generic features of the entanglement of link states in Chern-Simons theories with a compact group. In this discussion, we will conjecture a classification (albeit broad) of links via multi-partite entanglement of their corresponding states. We will end the chapter moving away from compact groups and discuss hyperbolic links in SL(2,C) Chern-Simons theory in the semi-classical limit. In particular, utilizing the generalized volume conjecture, we will show that the entanglement structure of hyperbolic links is greatly constrained in this semi-classical limit. Finally, in chapter 3 we switch gears and discuss renormalization of free bosonic field theory. The content of this chapter is drawn from the work of [4]. Motivated by work establishing duality between the singlet sector of O(N) vector models at large N and higher spin gauge theory on Anti de Sitter (AdS) space, we develop an exact renormalization group (ERG) framework for discussing the renormalization of a large class of O(N) singlet states. We will show that for the ground state and the excited states in this class the ERG is implemented by the action of unitary operators (that can be taken to be local). Consequently, the ERG framework can be regarded as a continuum tensor network. We contrast this tensor network with some well known tensor network renormalization schemes such as the muti-scale entanglement renormalization ansatz (MERA) and its proposed continuum version (cMERA). We also comment on the nature of this network with ordinary Wilsonian renormalization. One of the central features of the ERG network is how it acts on the momentum space entanglement of the field theory. In particular, we argue that for excited states, the ERG disentangles modes that lie just above and below the UV cutoff.