First law and quantum correction for holographic entanglement contour

Abstract

Entanglement entropy satisfies a first law-like relation, which equates the first order perturbation of the entanglement entropy for the region AA to the first order perturbation of the expectation value of the modular Hamiltonian, \delta S_{A}=\delta \langle K_A \rangleδSA=δ⟨KA⟩. 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})\rangleδsA(𝐱)=δ⟨kA(𝐱)⟩. Here the contour functions s_{A}(\textbf{x})sA(𝐱) and k_{A}(\textbf{x})kA(𝐱) capture the contribution from the degrees of freedom at \textbf{x}𝐱 to S_{A}SA and K_AKA respectively. In some simple cases k_{A}(\textbf{x})kA(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.