Bootstrapping Quantum Extremal Surfaces I: The Area Operator

TL;DR

This work develops a concrete bootstrap program linking quantum extremal surfaces to microscopic CFT data in AdS$_3$/CFT$_2$, by computing the CFT entanglement entropy for one-particle excited states and matching it to the bulk area operator up to order $O(c^{-1})$. The authors show that the Virasoro identity block is fully captured by the area operator at this order, while double-trace contributions require bulk entanglement entropy, thereby establishing a precise dictionary between CFT OPE data and geometric plus entanglement data in the bulk. They perform a detailed bulk EFT analysis, calculating the metric backreaction to second order and the resulting area variations, and demonstrate agreement with the CFT results for the leading $1/c$ corrections in the small-interval limit. The results validate a systematic program to derive quantum HRRT-type relations from microscopic CFT data and guide future work on bulk entanglement, graviton contributions, and extensions to more intricate entangling regions.

Abstract

Quantum extremal surfaces are central to the connection between quantum information theory and quantum gravity and they have played a prominent role in the recent progress on the information paradox. We initiate a program to systematically link these surfaces to the microscopic data of the dual conformal field theory, namely the scaling dimensions of local operators and their OPE coefficients. We consider CFT states obtained by acting on the vacuum with single-trace operators, which are dual to one-particle states of the bulk theory. Focusing on AdS$_3$/CFT$_2$, we compute the CFT entanglement entropy to second order in the large $c$ expansion where quantum extremality becomes important and match it to the expectation value of the bulk area operator. We show that to this order, the Virasoro identity block contributes solely to the area operator.

Figures (1)

• Figure 1: A fixed time slice of an asymptotically AdS$_3$ spacetime with classical extremal surface $\Sigma_{A}^{\mathrm{cl}}$ (red) given by the spacelike geodesic anchored on the boundary of the CFT interval $A$ (green). The quantum extremal surface $\Sigma_{A}$ (blue) is a spacelike curve that extremizes the generalized entropy and the (quantum) homology surface $R_{A}$ (orange) is the region bounded by $\Sigma_{A}$ and $A$.