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Entanglement entropies from entanglement contour: annuli and spherical shells

Muxin Han, Qiang Wen

Published May 14, 2019• 13 citations
OpenAlex

Abstract

The partial entanglement entropy $s_{\mathcal{A}}(\mathcal{A}_i)$ captures the contribution from the subset $\mathcal{A}_i$ of the region $\mathcal{A}$ to the total entanglement entropy $S_{\mathcal{A}}$ of $\mathcal{A}$. The partial entanglement entropy proposal \cite{Wen:2018whg,Wen:2019ubu} claims that $s_{\mathcal{A}}(\mathcal{A}_i)$ equals to a linear combination of the entanglement entropies of certain relevant subsets in $\mathcal{A}$. 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.

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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.
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.
This is an extended version of our short report hep-th/0603001, where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.
We study the structure and dynamics of entanglement in CFTs and black holes. We use a local entanglement measure, the entanglement contour, which is a spatial density function for von Neumann entropy with some additional properties. The entanglement contour can be calculated in many 1+1d condensed matter systems and simple models of black hole evaporation. We calculate the entanglement contour of a state excited by a splitting quench, and find universal results for the entanglement contours of low energy non-equilibrium states in 2d CFTs. We also calculate the contour of a non-gravitational bath coupled to an extremal AdS _2 2 black hole, and find that the contour only has finite support within the bath, due to an island phase transition. The particular entanglement contour proposal we use quantifies how well the bath’s state can be reconstructed from its marginals, through its connection to conditional mutual information, and the vanishing contour is a reflection of the protection of bulk island regions against erasures of the boundary state.
We study the structure and dynamics of entanglement in CFTs and black holes. We use a local entanglement measure, the entanglement contour, which is a spatial density function for von Neumann entropy with additional properties. We calculate the entanglement contour of a state excited by a splitting quench, and find universal results for the entanglement contours of low energy non-equilibrium states in 2d CFTs. We also calculate the contour of a non-gravitational bath coupled to an extremal AdS2_22​ black hole, and find that the contour only has finite support within the bath, due to an island phase transition. The particular entanglement contour proposal we use quantifies how well the bath's state can be reconstructed from its marginals, through its connection to conditional mutual information, and the vanishing contour is a reflection of the protection of bulk island regions against erasures of the boundary state.