Abstract The Ryu-Takayanagi prescription can be cast in terms of a set of microscopic threads that help visualize holographic entanglement in terms of distillation of EPR pairs. While this framework has been exploited for regions with a high degree of symmetry, we take the first steps towards understanding general entangling regions, focusing on AdS4. Inspired by simple constructions achieved for the case of disks and the half-plane, we reformulate bit threads in terms of a magnetic-like field generated by a current flowing through the boundary of the entangling region. The construction is possible for these highly symmetric settings, leading us to a modified Biot-Savart law in curved space that fully characterizes the entanglement structure of the state. For general entangling regions, the prescription breaks down as the corresponding modular Hamiltonians become inherently nonlocal. We develop a formalism for general shape deformations and derive a flow equation that accounts for these effects as a systematic expansion. We solve this equation for a complete set of small deformations and show that the structure of the expansion explicitly codifies the expected nonlocalities. Our findings are consistent with numerical results existing in the literature, and shed light on the fundamental nature of quantum entanglement as a nonlocal phenomenon.
Abstract The Ryu-Takayanagi prescription can be cast in terms of a set of microscopic threads that help visualize holographic entanglement in terms of distillation of EPR pairs. While this framework has been exploited for regions with a high degree of symmetry, we take the first steps towards understanding general entangling regions, focusing on AdS4. Inspired by simple constructions achieved for the case of disks and the half-plane, we reformulate bit threads in terms of a magnetic-like field generated by a current flowing through the boundary of the entangling region. The construction is possible for these highly symmetric settings, leading us to a modified Biot-Savart law in curved space that fully characterizes the entanglement structure of the state. For general entangling regions, the prescription breaks down as the corresponding modular Hamiltonians become inherently nonlocal. We develop a formalism for general shape deformations and derive a flow equation that accounts for these effects as a systematic expansion. We solve this equation for a complete set of small deformations and show that the structure of the expansion explicitly codifies the expected nonlocalities. Our findings are consistent with numerical results existing in the literature, and shed light on the fundamental nature of quantum entanglement as a nonlocal phenomenon.
This paper systematically develops the concept of entanglement threads that characterize the entanglement structure of holographic duality. Behind this framework lies a simple philosophy: holographic quantum entanglement can be visualized using threadlike objects. Inspired by the fact that tensor network models can be deformed into a quantum circuit form with flow-conserving features, we abstract the concept of entanglement threads. These entanglement threads can be understood as a preset ensemble of wires in a holographic quantum circuit, and we propose that they characterize the underlying partially ordered structure of holographic quantum entanglement. Combining the concepts of entanglement threads and kinematic space, a elegant circuit interpretation for the holographic complexity is provided. We also clarify the connection and distinction between entanglement threads and the previously proposed concept of bit threads.
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.
Nonlocal interactions are known to generate volume-law entanglement entropy. However, their deeper impact on the fine structure of quantum correlations remains a key open question. In this work, we explore a bosonic nonlocal field theory, examining correlation measures beyond entanglement entropy, namely, mutual information and tripartite information. Using numerical lattice simulations, we show that the nonlocality scale, A, not only determines the onset of volume-law behavior but also leads to striking features: notably, extremely long-range mutual information and an unusual monogamy structure. In this regime, increasing the separation between large regions can paradoxically enhance their multipartite entanglement. Through holographic duality, we verify that the Ryu-Takayanagi formula correctly captures the volume-law scaling of entropy. Yet, a significant tension emerges: while the field theory reveals rich spatial correlations, the holographic model predicts a complete suppression of both mutual and tripartite information in the volume-law phase. This non-monogamous behavior in the holographic description stands in sharp contrast to the monogamous and highly structured entanglement observed in the field theory. Our results demonstrate that nonlocality gives rise to quantum states of such complexity that conventional geometric models of spacetime fall short. This points to the need for a new framework that goes beyond geometry to fully capture the nature of these correlations.
A bstract We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions.
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 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.