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.

中文翻译：

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全息纠缠轮廓的第一定律和量子校正

纠缠熵满足一阶类律关系，即$A$区域的纠缠熵的一阶扰动等于模哈密顿量的期望值的一阶扰动，$\delta S_{A}=\delta \langle K_A \rangle$. 我们建议这种关系有一个更精细的版本，即纠缠轮廓的一阶扰动等于模哈密顿量的轮廓的一阶扰动，即 $\delta s_{A}(\textbf{x}) =\delta \langle k_{A}(\textbf{x})\rangle$. 这里的轮廓函数 $s_{A}(\textbf{x})$ 和 $k_{A}(\textbf{x})$ 捕获了 $\textbf{x}$ 处的自由度对 $S_ 的贡献{A}$ 和 $K_A$ 分别。在一些简单的情况下，$k_{A}(\textbf{x})$ 由应力张量决定。我们还分别使用纠缠楔的精细结构和部分纠缠熵 (PEE) 的加性线性组合 (ALC) 建议来评估对纠缠轮廓的量子校正。精细结构图表明，对边界PEE的量子校正可以识别为某个块体区域的块体PEE。而\textit{ALC 提议} 表明对边界PEE 的量子校正来自体纠缠熵的线性组合。我们专注于具有局部模块化哈密顿量的全息理论和应用 \textit{ALC 提议} 的量子场论配置。精细结构图表明，对边界PEE的量子校正可以识别为某个块体区域的块体PEE。而\textit{ALC 提议} 表明对边界PEE 的量子校正来自体纠缠熵的线性组合。我们专注于具有局部模块化哈密顿量的全息理论和应用 \textit{ALC 提议} 的量子场论配置。精细结构图表明，对边界PEE的量子校正可以识别为某个块体区域的块体PEE。而\textit{ALC 提议} 表明对边界PEE 的量子校正来自体纠缠熵的线性组合。我们专注于具有局部模块化哈密顿量的全息理论和应用 \textit{ALC 提议} 的量子场论配置。