First law and quantum correction for holographic entanglement contour

<jats:p>Entanglement entropy satisfies a first law-like relation, which equates the first order perturbation of the entanglement entropy for the region <jats:inline-formula><jats:alternatives><jats:tex-math>A</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi></mml:math></jats:alternatives></jats:inline-formula> to the first order perturbation of the expectation value of the modular Hamiltonian, <jats:inline-formula><jats:alternatives><jats:tex-math>\delta S_{A}=\delta \langle K_A \rangle</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy="false" form="prefix">⟨</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy="false" form="postfix">⟩</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. 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. <jats:inline-formula><jats:alternatives><jats:tex-math>\delta s_{A}(\textbf{x})=\delta \langle k_{A}(\textbf{x})\rangle</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>s</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mtext mathvariant="bold">𝐱</mml:mtext><mml:mo stretchy="false" form="postfix">)</mml:mo><mml:mo>=</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy="false" form="prefix">⟨</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mtext mathvariant="bold">𝐱</mml:mtext><mml:mo stretchy="false" form="postfix">)</mml:mo><mml:mo stretchy="false" form="postfix">⟩</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. Here the contour functions <jats:inline-formula><jats:alternatives><jats:tex-math>s_{A}(\textbf{x})</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mtext mathvariant="bold">𝐱</mml:mtext><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>k_{A}(\textbf{x})</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mtext mathvariant="bold">𝐱</mml:mtext><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> capture the contribution from the degrees of freedom at <jats:inline-formula><jats:alternatives><jats:tex-math>\textbf{x}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mtext mathvariant="bold">𝐱</mml:mtext></mml:math></jats:alternatives></jats:inline-formula> to <jats:inline-formula><jats:alternatives><jats:tex-math>S_{A}</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>S</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>K_A</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>K</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula> respectively. In some simple cases <jats:inline-formula><jats:alternatives><jats:tex-math>k_{A}(\textbf{x})</jats:tex-math><mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mtext mathvariant="bold">𝐱</mml:mtext><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> 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. </jats:p>