Entanglement entropy satisfies a first law-like relation, which equates the first order perturbation of the entanglement entropy for the region A A to the first order perturbation of the expectation value of the modular Hamiltonian, \delta S_{A}=\delta \langle K_A \rangle δ S A = δ ⟨ K A ⟩ . We propose that this relation has a finer version which states that, the first order perturbation of the entanglement contour equals to the first order perturbation of the contour of the modular Hamiltonian, i.e. \delta s_{A}(\textbf{x})=\delta \langle k_{A}(\textbf{x})\rangle δ s A ( 𝐱 ) = δ ⟨ k A ( 𝐱 ) ⟩ . Here the contour functions s_{A}(\textbf{x}) s A ( 𝐱 ) and k_{A}(\textbf{x}) k A ( 𝐱 ) capture the contribution from the degrees of freedom at \textbf{x} 𝐱 to S_{A} S A and K_A K A respectively. In some simple cases k_{A}(\textbf{x}) k A ( 𝐱 ) is determined by the stress tensor. We also evaluate the quantum correction to the entanglement contour using the fine structure of the entanglement wedge and the additive linear combination (ALC) proposal for partial entanglement entropy (PEE) respectively. The fine structure picture shows that, the quantum correction to the boundary PEE can be identified as a bulk PEE of certain bulk region. While the shows that the quantum correction to the boundary PEE comes from the linear combination of bulk entanglement entropy. We focus on holographic theories with local modular Hamiltonian and configurations of quantum field theories where the applies.
Entanglement entropy satisfies a first law-like relation, which equates the first order perturbation of the entanglement entropy for the region A A to the first order perturbation of the expectation value of the modular Hamiltonian, \delta S_{A}=\delta \langle K_A \rangle δ S A = δ ⟨ K A ⟩ . We propose that this relation has a finer version which states that, the first order perturbation of the entanglement contour equals to the first order perturbation of the contour of the modular Hamiltonian, i.e. \delta s_{A}(\textbf{x})=\delta \langle k_{A}(\textbf{x})\rangle δ s A ( 𝐱 ) = δ ⟨ k A ( 𝐱 ) ⟩ . Here the contour functions s_{A}(\textbf{x}) s A ( 𝐱 ) and k_{A}(\textbf{x}) k A ( 𝐱 ) capture the contribution from the degrees of freedom at \textbf{x} 𝐱 to S_{A} S A and K_A K A respectively. In some simple cases k_{A}(\textbf{x}) k A ( 𝐱 ) is determined by the stress tensor. We also evaluate the quantum correction to the entanglement contour using the fine structure of the entanglement wedge and the additive linear combination (ALC) proposal for partial entanglement entropy (PEE) respectively. The fine structure picture shows that, the quantum correction to the boundary PEE can be identified as a bulk PEE of certain bulk region. While the shows that the quantum correction to the boundary PEE comes from the linear combination of bulk entanglement entropy. We focus on holographic theories with local modular Hamiltonian and configurations of quantum field theories where the applies.
Entanglement entropy satisfies a first law-like relation, which equates the first order perturbation of the entanglement entropy for the region A to the first order perturbation of the expectation value of the modular Hamiltonian, δSA=δ⟨KA⟩. We propose that this relation has a finer version which states that, the first order perturbation of the entanglement contour equals to the first order perturbation of the contour of the modular Hamiltonian, i.e. δsA(x)=δ⟨kA(x)⟩. Here the contour functions sA(x) and kA(x) capture the contribution from the degrees of freedom at x to SA and KA respectively. In some simple cases kA(x) is determined by the stress tensor. We also evaluate the quantum correction to the entanglement contour using the fine structure of the entanglement wedge and the additive linear combination (ALC) proposal for partial entanglement entropy (PEE) respectively. The fine structure picture shows that, the quantum correction to the boundary PEE can be identified as a bulk PEE of certain bulk region. While the \textit{ALC proposal} shows that the quantum correction to the boundary PEE comes from the linear combination of bulk entanglement entropy. We focus on holographic theories with local modular Hamiltonian and configurations of quantum field theories where the \textit{ALC proposal} applies.
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.
Entanglement contour characterizes the spatial structure of entanglement and quantifies the contribution from the degrees of freedom in any subset of the region A to the total entanglement entropy SA. Recently in \cite{Wen:2018whg} the author gave a simple proposal for the entanglement contour (or partial entanglement entropy) in two-dimensional theories, which involves the entanglement entropies of all the subsets inside A. In this paper we explicitly study this proposal and show it satisfies many rational requirements for entanglement contour. Together with the holographic picture constructed with the modular planes \cite{Wen:2018whg}, we propose the correspondence between bulk geodesic chords and boundary partial entanglement entropies, which can be considered as a finer version of the Ryu-Takayanagi (RT) formula. As an example we calculate the fine correspondence between the points on A and the points on the RT surface EA for the BTZ black hole. We also give a strategy to extract the local modular flow from our proposal.
We extend the holographic test of entanglement cohomology from d=2 to higher-dimensional conformal field theories. For a d-dimensional CFT with a spherical entangling region A, the vacuum modular Hamiltonian is known explicitly, and the boundary relative entropy of a primary-operator perturbation can be computed via conformal Ward identities. On the bulk side, the perturbation is dual to a scalar field in AdSd+1, and the bulk canonical energy is computed using the JLMS relation. We carry out the computation explicitly for d=3 and d=4 and find that the boundary and bulk results agree at first order in the perturbation, with identical coefficients κdΔ2/CT, where κ3=π2/12 and κ4=3π3/16. The result extends the quantitative verification of the holographic dual of entanglement cohomology beyond the d=2 case treated in the companion paper and confirms that the modular Hamiltonian perturbation is the correct holographic image of the first entanglement cohomology class. We discuss the role of the conformal anomaly, the shape of the entanglement wedge, and an outlook on TsT-deformed backgrounds. \keywords{Entanglement entropy \and Cyclic cohomology \and von Neumann algebras \and Holography \and Modular Hamiltonian}
For holographic CFT states near the vacuum, entanglement entropies for spatial subsystems can be expressed perturbatively as an expansion in the one-point functions of local operators dual to light bulk fields. Using the connection between quantum Fisher information for CFT states and canonical energy for the dual spacetimes, we describe a general formula for this expansion up to second-order in the one-point functions, for an arbitrary ball-shaped region, extending the first-order result given by the entanglement first law. For two-dimensional CFTs, we use this to derive a completely explicit formula for the second-order contribution to the entanglement entropy from the stress tensor. We show that this stress tensor formula can be reproduced by a direct CFT calculation for states related to the vacuum by a local conformal transformation. This result can also be reproduced via the perturbative solution to a non-linear scalar wave equation on an auxiliary de Sitter spacetime, extending the first-order result in arXiv:1509.00113 .
Abstract The Entanglement contour function quantifies the contribution from each degree of freedom in a region A A to the entanglement entropy SA S A . Recently in [1] the author gave two proposals for the entanglement contour in two-dimensional theories. The first proposal is a fine structure analysis of the entanglement wedge, which applies to holographic theories. The second proposal is a claim that for general two-dimensional theories the partial entanglement entropy is given by a linear combination of entanglement entropies of relevant subsets inside A A . In this paper, we further study the partial entanglement entropy proposal by showing that it satisfies all the rational requirements proposed previously. We also extend the fine structure analysis from vacuum AdS space to BTZ black holes. Furthermore, we give a simple prescription to generate the local modular flows for two-dimensional theories from only the entanglement entropies without refer to the explicit Rindler transformations.