Entanglement contour and modular flow from subset entanglement entropies

Abstract

The Entanglement contour function quantifies the contribution from each degree of freedom in a region $\mathcal{A}$ to the entanglement entropy $S_{\mathcal{A}}$. Recently in \cite{Wen:2018whg} 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 $\mathcal{A}$. 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.

v2,9pages,8figures,version improved; v3,22 pages, name changed, clarification of the modular plane added, fine structure analysis for the BTZ background added, more discussions on the modular flows; v4, matching with the published version on JHEP

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• Formulas for Partial Entanglement Entropy
• The PEE aspects of entanglement islands from bit threads
• Local measures of entanglement in black holes and CFTs
• Deriving the PEE proposal from the Locking bit thread configuration
• Covariant entanglement wedge cross-section, balanced partial entanglement and gravitational anomalies
• Ownerless island and partial entanglement entropy in island phases
• Geometrizing the Partial Entanglement Entropy: from PEE Threads to Bit Threads
• On the entanglement contour of excited states in the holographic CFT
• Entanglement structures from modified IR geometry
• Weaving the (AdS) spaces with partial entanglement entropy threads
• First Law and Quantum Correction for Holographic Entanglement Contour
• Holographic Tensor Networks as Tessellations of Geometry
• A contour for the entanglement negativity of bosonic Gaussian states