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arXiv:1907.08126

Lectures on entanglement entropy in field theory and holography

Matthew Headrick

Published Jul 18, 2019• 33 citations
arXivHTMLPDF

Abstract

These notes, based on lectures given at various schools over the last few years, aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how the Ryu-Takayanagi formula geometrically realizes general features of field-theory entanglement, while revealing special properties of holographic theories. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.

DOI
10.48550/arxiv.1907.08126

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These notes, based on lectures given at various schools over the last few years, aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how the Ryu-Takayanagi formula geometrically realizes general features of field-theory entanglement, while revealing special properties of holographic theories. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.
These lecture notes aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how that physics is realized geometrically by the Ryu-Takayanagi formula. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.
These lecture notes aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how that physics is realized geometrically by the Ryu-Takayanagi formula. In order to make the notes somewhat self-contained for readers whose background is in high-energy theory, a brief introduction to the relevant aspects of quantum information theory is included.
The purpose of these lecture notes is to give an introduction to entanglement entropy with an emphasis on holography. Apart from the standard definition and properties of entanglement entropy, we explicitly derive the relation between the coefficients of the conformal anomaly and the universal term in the entropy of a d-dimensional CFT. We also discuss the prescription for calculating the entanglement entropy holographically in theories dual to Einstein-Hilbert and Lovelock gravity and present several examples in detail.
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.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.