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arXiv:1902.06905 · Journal of High Energy Physics

Entanglement contour and modular flow from subset entanglement entropies

Qiang Wen

Published May 1, 2020• 45 citations
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Abstract

Abstract The Entanglement contour function quantifies the contribution from each degree of freedom in a region $$ \mathcal{A} $$ A to the entanglement entropy $$ {S}_{\mathcal{A}} $$ 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 $$ \mathcal{A} $$ 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.

DOI
10.1007/jhep05(2020)018

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Abstract The Entanglement contour function quantifies the contribution from each degree of freedom in a region A\mathcal{A}A A to the entanglement entropy SA{S}_{\mathcal{A}}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\mathcal{A}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.
Entanglement contour characterizes the spatial structure of entanglement and quantifies the contribution from the degrees of freedom in any subset of the region A\mathcal{A}A to the total entanglement entropy SAS_{\mathcal{A}}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\mathcal{A}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\mathcal{A}A and the points on the RT surface EA\mathcal{E}_{\mathcal{A}}EA​ for the BTZ black hole. We also give a strategy to extract the local modular flow from our proposal.
We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3/CFT2{\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}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\mathcal{A}A and the points on the Ryu-Takayanagi (RT) surface EA{\mathcal{E}}_{\mathcal{A}}EA​. In the same sense an arbitrary subinterval A2{\mathcal{A}}_{2}A2​ of A\mathcal{A}A will correspond to a subinterval E2{\mathcal{E}}_{2}E2​ of EA{\mathcal{E}}_{\mathcal{A}}EA​. This fine correspondence indicates that the length of E2{\mathcal{E}}_{2}E2​ captures the contribution sA(A2){s}_{\mathcal{A}}({\mathcal{A}}_{2})sA​(A2​) from A2{\mathcal{A}}_{2}A2​ to the entanglement entropy SA{S}_{\mathcal{A}}SA​, hence gives the contour function for entanglement entropy. Furthermore we propose that sA(A2){s}_{\mathcal{A}}({\mathcal{A}}_{2})sA​(A2​) in general can be written as a simple linear combination of entanglement entropies of single intervals inside A\mathcal{A}A. This proposal passes several nontrivial tests.
The partial entanglement entropy sA(Ai)s_{\mathcal{A}}(\mathcal{A}_i)sA​(Ai​) captures the contribution from the subset Ai\mathcal{A}_iAi​ of the region A\mathcal{A}A to the total entanglement entropy SAS_{\mathcal{A}}SA​ of A\mathcal{A}A. The partial entanglement entropy proposal \cite{Wen:2018whg,Wen:2019ubu} claims that sA(Ai)s_{\mathcal{A}}(\mathcal{A}_i)sA​(Ai​) equals to a linear combination of the entanglement entropies of certain relevant subsets in A\mathcal{A}A. We derive the differential version of this proposal which can directly generate the scheme independent entanglement contour function. Furthermore we derived the sufficient condition to apply the differential version of proposal. On the other way around, the proposal indicates that given the partial entanglement entropy we can calculate the entanglement entropies for the subsets. Following this idea, on the field theory side we analytically calculated the entanglement contour functions and entanglement entropies for annuli and spherical shells in holographic CFTs in arbitrary dimensions. We also comment on the phase transition of the mutual information, which is calculated holographically, across an annulus.
We study the entanglement contour and partial entanglement entropy (PEE) in quantum field theories in 3 and higher dimensions. The entanglement entropy is evaluated from a certain limit of the PEE with a geometric regulator. In the context of the entanglement contour, we classify the geometric regulators, study their difference from the UV regulators. Furthermore, for spherical regions in conformal field theories (CFTs) we find the exact relation between the UV and geometric cutoff, which clarifies some subtle points in the previous literature. We clarify a subtle point of the additive linear combination (ALC) proposal for PEE in higher dimensions. The subset entanglement entropies in the ALC proposal should all be evaluated as a limit of the PEE while excluding a fixed class of local-short-distance correlation. Unlike the 2-dimensional configurations, naively plugging the entanglement entropy calculated with a UV cutoff will spoil the validity of the ALC proposal. We derive the entanglement contour function for spherical regions, annuli and spherical shells in the vacuum state of general-dimensional CFTs on a hyperplane.
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.
We explore the fine structure of the holographic entanglement entropy proposal (the Ryu-Takayanagi formula) in AdS3_33​/CFT2_{2}2​. 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\mathcal{A}A and the points on the Ryu-Takayanagi (RT) surface EA\mathcal{E}_{\mathcal{A}}EA​. In the same sense an arbitrary subinterval A2\mathcal{A}_2A2​ of A\mathcal{A}A will correspond to a subinterval E2\mathcal{E}_2E2​ of EA\mathcal{E}_{\mathcal{A}}EA​. We propose that the length of E2\mathcal{E}_2E2​ captures the contribution from A2\mathcal{A}_2A2​ to the entanglement entropy SAS_{\mathcal{A}}SA​, and compare it with the mutual information between A2\mathcal{A}_2A2​ and the complement Ac\mathcal{A}^cAc.