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.
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.
The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $\mathrm{\ensuremath{\Sigma}}$ with boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region A and its complement A are specified on the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$, we show that the boundary entanglement entropy S(A) of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., S(A) is proportional to the minimal area of the bulk surface attached to the boundary of A in $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$.
We study the holographic properties of a class of quantum geometry states characterized by a superposition of discrete geometric data, in the form of generalized tensor networks. This class specifically includes spin networks, the kinematic states of lattice gauge theory, and discrete quantum gravity. We employ an algebraic, operatorial definition of holography based on quantum information channels, an approach which is particularly valuable in settings, such as the one we consider, where the relevant Hilbert space of states does not factorize into subsystem Hilbert spaces due to gauge invariance. We apply random tensor network techniques (successfully used in the AdS/CFT context) to analyze information transport properties of the bulk-to-boundary and boundary-to-boundary maps associated with this superposition of quantum geometries and produce typicality results about the average over the geometric data coloring the fixed graph structure. In this context, one naturally obtains a nontrivial area operator encoding the dominant contribution to entropy calculations. Among our main results is the requirement that one can only isometrically map a bulk region onto boundaries with fixed total area. We furthermore inquire about similar state-induced mappings between segments of the boundary and discuss related conditions for isometric behavior. These generalizations make further steps toward quantum gravity implementations of tensor network holography.
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.
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
For quantum gravity states associated with open spin network graphs, we study how the entanglement entropy of the boundary degrees of freedom (spins on open edges) is affected by the bulk data, specifically by its combinatorial structure and by the quantum correlations among intertwiner degrees of freedom. For a specific assignment of bulk edge spins and slightly entangled intertwiners, we recover the Ryu-Takayanagi formula (with a properly discrete geometric notion of area, thanks to the underlying quantum gravity formalism) and its corrections due to the entanglement entropy of the bulk state. We also show that the presence of a region with highly entangled intertwiners deforms the minimal-area surface, which is then prevented from entering that region when the entanglement entropy of the latter exceeds a certain bound. This entanglement-based mechanism leads thus to the formation of a black hole--like region in the bulk.
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.