Ryu–Takayanagi formula

Remarks

Ryu and Takayanagi proposed the formula in 2006, modelled on the Bekenstein–Hawking entropy of a horizon. Lewkowycz and Maldacena (2013) derived it within AdS/CFT by a gravitational version of the replica trick, which computes $\tr \rho_A^n$ from bulk geometries and takes $n \to 1$. Example: for $d = 2$ and an interval of length $\ell$ in the vacuum, $\gamma_A$ is a geodesic in $\mathrm{AdS}_3$ of radius $L$ with regularized length $2L \log(\ell/\epsilon)$, where $\epsilon$ is a cutoff. With the Brown–Henneaux central charge $c = 3L/(2G_N)$ this gives $S_A = \frac{c}{3} \log(\ell/\epsilon)$, the known entanglement entropy of an interval in a two-dimensional conformal field theory. For a time-dependent state, the minimal surface on a slice is replaced by an extremal surface in the full spacetime. This is the covariant formula of Hubeny, Rangamani and Takayanagi (2007). Quantum corrections add the entropy of bulk fields between $A$ and $\gamma_A$.

Going deeper

For a student

In AdS/CFT the boundary entanglement entropy is hard to compute directly, but the right-hand side is a classical minimal-surface problem in the bulk. The formula implies inequalities such as strong subadditivity and the monogamy of mutual information, which can be proved geometrically by cutting and gluing minimal surfaces. When the bulk contains a black hole and $A$ is the whole boundary, $\gamma_A$ wraps the horizon and the formula reduces to the Bekenstein–Hawking entropy of the thermal state. It is the starting point of entanglement wedge reconstruction and of the quantum extremal surface prescription used for the black hole information problem.