Quantum extremal surface

A quantum extremal surface is a generalization of the classical extremal surface in quantum gravity, used to calculate the entanglement entropy of subregions in theories with gravitational dynamics. In the context of the holographic principle and AdS/CFT correspondence, the quantum extremal surface formula gives the correct entropy of Hawking radiation, reproducing the Page curve and resolving the black hole information paradox in a way that purely classical geometry cannot.

The key insight is that when quantum effects are included, the entropy of a region is not simply the area of the minimal surface (as in the classical Ryu-Takayanagi formula). Instead, one must extremize a quantity that includes both the area and the bulk entropy of fields outside the surface. The quantum extremal surface can jump discontinuously as the region grows, leading to the phase transition that reproduces the Page curve. This demonstrates that quantum gravity requires a new notion of geometric entropy — one in which spacetime geometry and quantum entanglement are inseparable.