We propose a framework for preparing quantum states with a holographic entanglement structure, in the sense that the entanglement entropies are governed by minimal surfaces in a chosen bulk geometry. We refer to such entropies as holographic because they obey a relation between entropies and bulk minimal surfaces, known as the Ryu-Takayanagi (RT) formula, that is a key feature of holographic models of quantum gravity. Typically in such models, the bulk geometry is determined by solving Einstein's equations. Here, we simply choose a bulk geometry, then discretize the geometry into a coupling graph comprising bulk and boundary nodes. Evolving under this graph of interactions and measuring the bulk nodes leaves behind the desired pure state on the boundary. We numerically demonstrate that the resulting entanglement properties approximately reproduce the predictions of the Ryu-Takayanagi formula in the chosen bulk geometry. We consider graphs associated with hyperbolic disk and wormhole geometries, but the approach is general. We also introduce a procedure for decorating the graph which, when applied to the hyperbolic disk, yields a genuine CFT ground state with power-law correlations but which only partially obeys RT. The minimal ingredients in our proposal involve only Gaussian operations and measurements and are readily implementable in photonic and cold-atom platforms.
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5 pages + references + supplement, 4 figures in main text; 6 min video abstract available at https://www.youtube.com/watch?v=wVckaCwgfSU
We propose a framework for preparing quantum states with a holographic entanglement structure, in the sense that the entanglement entropies are governed by minimal surfaces in a chosen bulk geometry. We refer to such entropies as holographic because they obey a relation between entropies and bulk minimal surfaces, known as the Ryu-Takayanagi (RT) formula, that is a key feature of holographic models of quantum gravity. Typically in such models, the bulk geometry is determined by solving Einstein's equations. Here, we simply choose a bulk geometry, then discretize the geometry into a coupling graph comprising bulk and boundary nodes. Evolving under this graph of interactions and measuring the bulk nodes leaves behind the desired pure state on the boundary. We numerically demonstrate that the resulting entanglement properties approximately reproduce the predictions of the Ryu-Takayanagi formula in the chosen bulk geometry. We consider graphs associated with hyperbolic disk and wormhole geometries, but the approach is general. We also introduce a procedure for decorating the graph which, when applied to the hyperbolic disk, yields a genuine CFT ground state with power-law correlations but which only partially obeys RT. The minimal ingredients in our proposal involve only Gaussian operations and measurements and are readily implementable in photonic and cold-atom platforms.
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.
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.
We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3/CFT2. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are co-dimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A and the points on the Ryu-Takayanagi (RT) surface EA. In the same sense an arbitrary subinterval A2 of A will correspond to a subinterval E2 of EA. We propose that the length of E2 captures the contribution from A2 to the entanglement entropy SA, and compare it with the mutual information between A2 and the complement Ac.
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 explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3/CFT2. With the guidance from the boundary and bulk modular flows we find a natural slicing of the entanglement wedge with the modular planes, which are codimension one bulk surfaces tangent to the modular flow everywhere. This gives an one-to-one correspondence between the points on the boundary interval A and the points on the Ryu-Takayanagi (RT) surface EA. In the same sense an arbitrary subinterval A2 of A will correspond to a subinterval E2 of EA. This fine correspondence indicates that the length of E2 captures the contribution sA(A2) from A2 to the entanglement entropy SA, hence gives the contour function for entanglement entropy. Furthermore we propose that sA(A2) in general can be written as a simple linear combination of entanglement entropies of single intervals inside A. This proposal passes several nontrivial tests.
We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.