The Ryu-Takayanagi Formula from Quantum Error Correction

TL;DR

The paper reframes AdS/CFT through quantum error correction, deriving the Ryu-Takayanagi formula as a sequence of theorems that progressively generalize from conventional to subsystem to operator-algebra quantum error correction. It provides a boundary-centric, algebraic formulation using von Neumann algebras and a new notion of entropy on algebras, tying the RT area term to edge modes and central elements. The main contributions include equivalent-condition theorems for erasure correction, symmetrical RT relations in subsystem and algebraic codes, and an algebraic reconstruction theorem that extends holographic duality beyond simple tensor-factorized codes. This framework unifies subregion duality, entanglement wedge reconstruction, and bit-thread interpretations, while highlighting gauge constraints and edge-mode physics as central to the holographic entropy balance. The results offer a boundary-grounded, information-theoretic perspective on holography with clear implications for how bulk and boundary degrees of freedom encode one another.

Abstract

I argue that a version of the quantum-corrected Ryu-Takayanagi formula holds in any quantum error-correcting code. I present this result as a series of theorems of increasing generality, with the final statement expressed in the language of operator-algebra quantum error correction. In AdS/CFT this gives a "purely boundary" interpretation of the formula. I also extend a recent theorem, which established entanglement-wedge reconstruction in AdS/CFT, when interpreted as a subsystem code, to the more general, and I argue more physical, case of subalgebra codes. For completeness, I include a self-contained presentation of the theory of von Neumann algebras on finite-dimensional Hilbert spaces, as well as the algebraic definition of entropy. The results confirm a close relationship between bulk gauge transformations, edge-modes/soft-hair on black holes, and the Ryu-Takayanagi formula. They also suggest a new perspective on the homology constraint, which basically is to get rid of it in a way that preserves the validity of the formula, but which removes any tension with the linearity of quantum mechanics. Moreover they suggest a boundary interpretation of the "bit threads" recently introduced by Freedman and Headrick.

Figures (10)

• Figure 2: A holographic encoding circuit. $A$ is a CFT subregion, $\overline{A}$ is its complement, and $\mathcal{E}_A$ and $\mathcal{E}_{\overline{A}}$ are the bulk degrees of freedom in their respective entanglement wedges. We encode these bulk degrees of freedom into the CFT by acting with unitary transformations $U_A$ and $U_{\overline{A}}$ that mix $\mathcal{E}_A$ and $\mathcal{E}_{\overline{A}}$ with complementary pieces of a fixed state $|\chi\rangle$, which accounts for the remaining CFT degrees of freedom in $A$ and $\overline{A}$. The entanglement in the state $|\chi\rangle$ is the source of the area terms of the RT formulae for $S_A$ and $S_{\overline{A}}$, while the states that are fed into $\mathcal{E}_A$ and $\mathcal{E}_{\overline{A}}$ give the bulk entropy terms. We will see that nonvanishing entanglement in $|\chi\rangle$, and thus a nonvanishing area term, is necessary for the robust functioning of the code.
• Figure 3: Scalar lattice QED in 1+1 dimensions. Each spatial link gets an element of $U(1)$, and each internal site gets a complex scalar.
• Figure 4: An algebraic decomposition of the AdS-Schwarzschild geometry. $M_L$ lives in the blue region, $M_R=M_L'$ lives in the red region, and the center corresponds to edge modes on the bifurcation surface $\gamma$.
• Figure 5: Gravitational dressing in AdS. Truly local operators do not exist in gravitational theories, but we can define pseudo-local operators by shooting geodesics from the boundary Heemskerk:2012npKabat:2013wgaAlmheiri:2014lwaDonnelly:2015htaDonnelly:2015taaDonnelly:2016rvo. These operators will commute to all orders in perturbation theory with operators from which their entire geodesics are spacelike separated Almheiri:2014lwaDonnelly:2015hta, so provided that their geodesics lie entirely in $\mathcal{E}_A$ or $\mathcal{E}_{\overline{A}}$ then they will be in $M$ or $M'$ respectively, and they will commute with the area operator on the extremal surface (whose location is already defined gauge-invariantly without needing similar geodesics). The area operator is thus in the center $M\cap M'$.
• Figure 6: Freedman-Headrick threads and the circuit interpretation of the RT formula. The threads are shown in green in the left diagram; they are chosen to maximize the flux through $A$, and this maximal flux, determined by the bottleneck at $\gamma_A$, gives the entropy $S(\widetilde{\rho}_A)$. In the circuit diagram these threads correspond to the information flux through the state $|\chi\rangle$ that appears in eq. \ref{['sstate']}. We can thus interpret the Freedman-Headrick proposal as routing the circuit diagram through the bulk.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 3.1
• proof
• Theorem 4.1
• Theorem 5.1
• proof
• Proposition 6.1
• Definition A.1
• Definition A.2
• Definition A.3