Ryu-Takayanagi Prescription

Updated 27 November 2025

Ryu-Takayanagi Prescription is a geometric method that computes entanglement entropy by equating the area of a minimal (or extremal) surface to the entropy in holographic duals.
It extends to time-dependent scenarios through the HRT prescription and incorporates quantum corrections, tensor networks, and group field theory to refine its predictions.
This framework underlies key insights in Einstein gravity and quantum error correction, bridging geometric and algebraic perspectives in quantum field theories with gravitational duals.

The Ryu-Takayanagi (RT) prescription provides a geometric formula for computing the entanglement entropy of a spatial subregion in quantum field theories with gravitational holographic duals. In its original and most widely studied context—Einstein gravity in asymptotically AdS spacetimes—it equates the entanglement entropy of a boundary region to the area of a minimal bulk codimension-2 surface (the RT surface) homologous to the region, divided by 4 times Newton’s constant. Generalizations and reinterpretations have established the RT prescription as a universal principle rooted in bulk quantum error correction, operator algebraic structure, group field theory, and even abstract gravitational path integral formalism.

1. Minimal Surface Formula and Homology

The canonical RT formula applies to static bulk spacetimes with a time-reflection symmetry. For a CFT state with a semiclassical dual bulk geometry and boundary spatial region $A$ , one considers the set of bulk codimension-2 surfaces $m \sim A$ that are anchored on $\partial A$ and homologous to $A$ . Among these, the surface with minimal area, $m(A)$ , determines the entanglement entropy: $S(A) = \frac{\text{Area}[m(A)]}{4G_N}$ where $G_N$ is the bulk Newton constant. The homology condition requires that $m$ and $A$ together bound a bulk region $r$ [$1204.2309$].

Strong subadditivity (SSA) of entropy, a key property for physical entanglement measures, holds generally for the RT prescription so long as the bulk is static and smooth. It follows geometrically from the cut-and-paste operation on regions and their minimal surfaces, guaranteeing for adjacent intervals $A$ , $B$ , $C$ : $S(AB) + S(BC) \ge S(ABC) + S(B)$ and analogously for other combinations [$1204.2309$].

2. Covariant Extensions and Quantum Corrections

For non-static or time-dependent bulk spacetimes, the minimal surface is replaced by an extremal surface, leading to the Hubeny-Rangamani-Takayanagi (HRT) prescription: $S(A) = \frac{1}{4G_N} \min_{m \sim A,\, \delta\,\text{Area}[m]=0} \text{Area}[m]$ where the extremal condition is that the surface has vanishing mean curvature in the Lorentzian geometry [$1204.2309$].

Quantum corrections, higher-curvature effects, and non-Einstein dynamics are encompassed by the “quantum extremal surface” prescription. For region $A$ ,

$S(A) = \min_{\gamma\sim A} \, \underset{\mathrm{Ext}}{\mathrm{ext}}\,\Bigl[\,\frac{\mathrm{Area}(\gamma)}{4G_N} + S_{\rm bulk}(\Sigma)\,\Bigr]$

where $S_{\rm bulk}(\Sigma)$ is the von Neumann entropy of bulk quantum fields in the entanglement wedge bounded by $\gamma$ [$2405.14872$, $2508.14877$]. For actions with higher derivatives, the “area term” is replaced by Dong’s functional, involving an integral over the extrinsic curvature variations.

3. Emergence in Quantum Error-Correcting Codes

The RT prescription is a direct consequence of the quantum error-correcting structure of holographic codes. Any code with complementary recovery for boundary subalgebras associated to regions $A$ and $\bar{A}$ , and a bulk operator algebra with a nontrivial center (the “area operator”), admits the algebraic Ryu-Takayanagi formula: $S(\rho_A) = \langle A \rangle + S_{\rm bulk}(\rho)$ where the area operator $A$ is a central element associated to the entangling surface and $\langle A \rangle$ its expectation, and $S_{\rm bulk}(\rho)$ is the algebraic entropy on the bulk operator algebra. In operator-algebraic quantum error correction, these results persist in infinite-dimensional Hilbert spaces and Type I/II von Neumann factors [$1912.02240$, $2411.06361$].

In gauge theories or systems with edge modes, the area term is associated to superselection sectors at the entangling surface; the bulk term accounts for both quantum field entanglement and Shannon mixing entropy between superselection sectors [$1912.02240$].

4. Tensor Networks, Group Field Theory, and Minimal Cuts

Tensor-network models, specifically large-bond-dimension random tensor networks (RTN), explicitly reproduce the RT formula. For a network with bond dimension $D$ and boundary region $A$ ,

$S_{\rm EE}(A) \approx L_{\min}(A)\,\ln D$

where $L_{\min}(A)$ is the minimal number of links to sever to separate $A$ from its complement, corresponding to the minimal-cut surface in the bulk network graph. In models with gauge symmetry at each tensor, this result holds with the independent degrees of freedom appropriately reduced [$1711.09941$].

Group field theory (GFT) networks, generalizing spin networks of loop quantum gravity (LQG), map directly onto random tensor networks and support the RT law via replica-trick calculations of entanglement entropy. For free GFT with delta-function gluing, the entropy is proportional to the minimal number of links crossing the $A$ / $B$ cut, while with fixed-spin networks, the entropy is given by the sum of area eigenvalues across the minimal cut: $S_{\rm EE} = \sum_{\ell \in \partial AB} 2\pi\gamma j_\ell = \frac{\mathcal{A}_{\partial AB}}{4\ell_P^2}$ demonstrating the emergence of the geometric RT area law in background-independent quantum gravity formalisms [$1701.01383$, $1903.07344$].

Corrections from GFT interactions are subleading in the large- $D$ limit, leaving the leading RT prescription unaltered [$1903.07344$].

5. Boundary CFT Derivations and Statistical Mechanisms

In AdS $_3$ /CFT $_2$ , the RT formula emerges directly from Gaussian ensemble statistics of heavy-state OPE coefficients in a large- $c$ CFT. Gaussian Wick contractions dominate the calculation of multi-boundary wormhole states and their entanglement entropy. Each leading contraction corresponds to a distinct RT phase (homology class), and transitions between them realize replica-wormhole phase transitions. The statistical structure of the CFT encodes the geometric contractibility of replica cycles: $S_A = \min \Bigl\{\frac{L_A}{4G_N},\, \frac{L_B + L_C}{4G_N}\Bigr\}$ with $L_A$ the geodesic length homologous to $A$ [$2504.12388$].

For AdS $_2$ /CFT $_1$ , the entanglement entropy between boundaries is related to the logarithm of the regularized length of a closed geodesic near the rim of the Euclidean AdS $_2$ disk, matching the one-loop thermodynamic entropy and confirming the RT law in this dimension [$2502.01144$].

6. Generalized and Algebraic Frameworks

The RT prescription generalizes to arbitrary diffeomorphism-invariant quantum field theories via a functional integral over the theory's phase space. The replica trick applied to the reduced density matrix is realized as a path integral that, in the semiclassical limit, localizes to extremal codimension-2 surfaces with gravitational entropy generator $K[\mathcal{E}]$ : $S = \min_{\mathcal{E}_A \sim A} \frac{\text{Area}(\mathcal{E}_A)}{4G_N}$ with extensions to higher-derivative theories and inclusion of quantum corrections for the quantum extremal surface prescription [$2508.14877$].

Algebraically, the gravitational path integral, without reference to a boundary CFT, constructs bulk Hilbert spaces and algebras of observables, where the RT/HRT entanglement entropy is the von Neumann entropy of the left algebra acting on the gravitational state. This construction relies solely on basic axioms (finiteness, reflection positivity, factorization, continuity) and recovers RT/HRT in the semiclassical regime [$2310.02189$].

7. Extensions: Timelike, Confining, and Non-AdS Contexts

Recent work extends the RT prescription beyond spacelike regions. For example, timelike strip entanglement in static AdS black holes admits extremal surfaces traversing horizons, whose areas are reconstructed from data outside the horizon and causally related spacelike regions, implementing black hole complementarity [$2507.17805$, $2502.16774$].

In confining and soft-wall holographic models, the minimal surface prescription persists, and the entropy matches the field theory result across phase transitions and in asymptotic regimes [$2205.06724$]. Modified backgrounds or gauge symmetry can induce shifts in the structure of extremal surfaces, swallow-tail phase transitions, and subtle shifts in the Type I/II analytic continuation.

Summary Table: Core Elements of the Ryu-Takayanagi Prescription

Context	Surface Prescription	Quantum Corrections
Static AdS, Einstein gravity	$\min_{m\sim A}\mathrm{Area}(m)/4G_N$	None
Time-dependent/covariant (HRT)	$\mathrm{ext}_{m\sim A}\mathrm{Area}(m)/4G_N$	None
Quantum bulk fields	$\mathrm{ext}_{\gamma}\Big[\frac{\mathrm{Area}(\gamma)}{4G_N} + S_{\rm bulk}(\Sigma)\Big]$	$S_{\rm bulk}$
Gauge-invariant tensor/GFT networks	Minimal cut of links/edges, link area spectrum	$O(1/D)$ corrections
Diffeomorphism-invariant QFTs (generalized)	$\min_\mathcal{E} K[\mathcal{E}]$ (see above)	$S_{\rm bulk},$ higher-derivative