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.
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.
The AdS/CFT correspondence, a remarkable duality between certain gravitational theories in anti-de Sitter (AdS) spacetime and quantum field theories with conformal symmetry (CFT), has had a profound effect on the development of theoretical physics in the past two decades. Recently, many connections of AdS/CFT to quantum information theory have been found, in particular by providing gravitationally dual descriptions of various entanglement measures. Understanding these manifestions of AdS/CFT --- or more generally, the conjectured holographic principle encompassing it --- requires the combination of tools from both high-energy theory and quantum information physics. In this cumulative thesis, the convergence between these two fields is approached from two fronts: First, by calculations within the dual gravitational theory, and second, using a tensor network ansatz to describe the quantum states suspected to possess such a gravitational description. In the first approach, using the gravitational side of AdS/CFT, entanglement entropies of complicated 2+1-dimensional excited CFTs are computed, thus showing how the holographic approach provides access to systems previously out of reach of practical methods, while introducing new numerical methods that this approach necessitates. The second approach is given by tensor networks, a highly successful ansatz for computing properties of one- and two-dimensional quantum systems. Efficiently computable classes of tensor networks are tested in their ability to represent simple holographic systems, successfully reproducing both hyperbolic geometrical features as well as critical boundary states. In addition, the general properties of tensor networks on regular hyperbolic tesselations are considered, leading to new connections to models not previously considered in the context of holography. This interplay of different approaches to quantum information holography showcases the richness of this new field and suggests that a wide range of physical phenomena is accessible via the new tools now at our disposal.
It was found, by studying black holes, that the compatibility of the theory of gravity and the laws of quantum mechanics demands that the universe must act like a hologram. That is, all the information inside a region of the universe should be encoded on a so-called holographic screen of one-lesser dimension. This is termed the holographic principle. In a generic spacetime, such as an expanding universe, holographic screens depend on the choice of an observer, which is consistent with the notion that the observer is part of the system in cosmology. In the first part of this dissertation, we study the observer-dependence of holographic screens. This will help us understand how the fundamental description of the universe depends on the choice of the observer. Furthermore, we study the dynamics of the holographic screens and their geometry in the first part of this dissertation.An example where the holographic principle is manifest is the AdS-CFT correspondence. This correspondence connects quantum information quantities, like entanglement and complexity, to geometric quantities, like area and volume, respectively. In the second and the third part of this dissertation, we use the AdS-CFT correspondence as a tool to calculate the entanglement entropy and the computational complexity of a quenched CFT state. Studying the time evolution of the entanglement entropy following a quantum quench teaches us how a CFT state thermalizes. On the other hand, studying the time evolution of the computational complexity allows us to check the validity of various recent conjectures involving black holes and complexity.
In this thesis we review how entanglement entropy is related to the gravitational equations of motion by means of the AdS/CFT correspondence. We consider the entanglement entropy of a ball-shaped spatial region in the CFT vacuum and derive a holographic relation between the relative entropy and the equations of motion in the dual bulk spacetime. For small perturbations around the CFT vacuum, we show that the first law of entanglement entropy requires the dual metric to satisfy the linearized Einstein equation when the entanglement entropy is given by the area of an extremal surface in the bulk. We discuss a recent paper in which the complete Einstein equation is derived and explain why we believe this result is incomplete. This thesis serves both as a commentary on this paper, as well as a detailed review of the relationship between entanglement and gravity.
This dissertation reviews several recent advances at the intersection of quantum information and holography. In holography, properties of quantum systems admit a gravitational interpretation via the AdS/CFT correspondence. For holographic states, boundary entanglement entropy is dual to bulk geodesic areas, known as Ryu-Takayanagi surfaces. Furthermore, the viability to possess a holographic dual at all is constrained by entanglement structure. Accordingly, entanglement enables a coarse classification of states in a Hilbert space. Similarly, state transformation under operator groups also provides a classification on the Hilbert space. Stabilizer states, for example, are invariant under large sets of operations and consequently can be simulated on a classical computer. Cayley graphs offer a useful representation for a group of operators, where vertices represent group elements and edges represent generators. The orbit of a state under action of the group can also be represented as a "reachability graph", a quotient of the group Cayley graph. Reachability graphs can be dressed to encode entanglement information, making them a useful tool for studying entanglement dynamics. Quotienting a reachability graph by group elements that fix a state computable, e.g. entanglement entropy, builds a "contracted graph". Contracted graphs explicitly bound state parameter evolution in quantum circuits. In this thesis, an upper bound on entanglement entropy evolution in Clifford circuits is presented. Another important property of quantum systems is magic, which quantifies the difficulty of simulating a quantum state. Magic and entanglement play complementary roles when describing emergent phenomena in AdS/CFT. This work describes the interplay of entanglement and magic, offering holographic consequences for magic as cosmic brane back-reaction.
We give a pedagogical review of how concepts from quantum information theory build up the gravitational side of the anti-de Sitter/conformal field theory correspondence. The review is self-contained in that it only presupposes knowledge of quantum mechanics and general relativity; other tools-including holographic duality itself-are introduced in the text. We have aimed to give researchers interested in entering this field a working knowledge sufficient for initiating original projects. The review begins with the laws of black hole thermodynamics, which form the basis of this subject, then introduces the Ryu-Takayanagi proposal, the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) relation, and subregion duality. We discuss tensor networks as a visualization tool and analyze various network architectures in detail. Next, several modern concepts and techniques are discussed: Rényi entropies and the replica trick, differential entropy and kinematic space, modular Berry phases, modular minimal entropy, entanglement wedge cross-sections, bit threads, and others. We discuss the extent to which bulk geometries are fixed by boundary entanglement entropies, and analyze the relations such as the monogamy of mutual information, which boundary entanglement entropies must obey if a state has a semiclassical bulk dual. We close with a discussion of black holes, including holographic complexity, firewalls and the black hole information paradox, islands, and replica wormholes.
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.