# First Law and Quantum Correction for Holographic Entanglement Contour

### Submission summary

 As Contributors: Qiang Wen Preprint link: scipost_202106_00045v2 Date accepted: 2021-09-08 Date submitted: 2021-08-19 05:41 Submitted by: Wen, Qiang Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

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, $\delta S_{A}=\delta \langle K_A \rangle$. 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. $\delta s_{A}(\textbf{x})=\delta \langle k_{A}(\textbf{x})\rangle$. Here the contour functions $s_{A}(\textbf{x})$ and $k_{A}(\textbf{x})$ capture the contribution from the degrees of freedom at $\textbf{x}$ to $S_{A}$ and $K_A$ respectively. In some simple cases $k_{A}(\textbf{x})$ 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.

Published as SciPost Phys. 11, 058 (2021)

We thank the referees for recommendation and a careful reading our manuscript. Also thanks very much for the valuable comments and suggestions. In the revised manuscript, we made the corrections for further clarification following the referee's comments.

### List of changes

We improved the presentation of the manuscript, fixed the typos and grammatical errors we can find.

we add 4 references. These includes the interpretation of the ALC proposal as a conditional mutual information (page4) and mention that the entanglement contour has been applied to the island formla to depict the entanglement structure of the hawking radiation (page3).

we added the footnote 1, 5, 7 in accordance with the comments or suggestions from the referees.

we added a paragraph in the end of section 2 to explain a concern from both of the referees.

### Submission & Refereeing History

Resubmission scipost_202106_00045v2 on 19 August 2021
Submission scipost_202106_00045v1 on 28 June 2021

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