Quantum Extremal Surfaces

Definition

The quantum extremal surface (QES) prescription generalises the Ryu-Takayanagi / HRT formula by replacing the area functional with the generalised entropy: $S(A)=\min {\text{ext}}_{\gamma }[\text{Area}(\gamma )/(4G_N)+S_{\text{bulk}}({\Sigma }_{\gamma })]$, where ${\Sigma }_{\gamma }$ is the bulk region between $A$ and $\gamma$. Bulk-matter entanglement entropy enters on equal footing with area. The QES gives the island formula in the presence of a bath.

Role in Research

The QES prescription enters the holography course with the entanglement wedges of Lecture 19 and is used for the island of Lecture 26. It provides the mechanism by which the page-curve is reproduced from semiclassical gravity, via islands that appear at late times and cause the entropy to decrease.

Relations

• ryu-takayanagi-formula — the QES generalises the RT/HRT formula by including bulk-matter entropy.
• page-curve — the QES / island prescription is the tool that reproduces the Page curve.
• replica-trick-gravity — replica wormholes provide the path-integral derivation of the QES prescription.

Papers

• Engelhardt–Wall 2014, arXiv:1408.3203.
• Penington 2019, arXiv:1905.08255.
• Almheiri–Engelhardt–Marolf–Maxfield 2019, arXiv:1905.08762.

Notes

Course coverage: Lecture 19, Lecture 25 and Lecture 26.