Asymptotically isometric codes for holography

TL;DR

This work develops an algebraic QFT framework for holography by constructing single-trace Weyl algebras and large-$N$ nets that encode bulk physics in a boundary theory. It introduces asymptotically isometric codes $(\mathcal{C},V_N,\gamma_N)$, which are non-isometric at finite $N$ but become isometric in the $N\to\infty$ limit on fixed code states, enabling robust entanglement-wedge reconstructions. A central result is an asymptotic information-disturbance tradeoff that preserves boundary causality via net extensions, together with JLMS-type statements connecting bulk and boundary modular flow and relative entropy in the large-$N$ regime. The paper also discusses the hyperfinite condition, possible sector-dependent extensions, and how phase transitions and sector structure influence entanglement wedges, offering a principled route to model gravity as an approximate, sector-sensitive code with rich von Neumann algebraic structure. These insights help reconcile gravity’s locality and holographic bounds within a controlled, operator-algebraic QEC framework, with implications for entanglement entropy, bulk reconstruction, and phase structure in holographic theories.

Abstract

The holographic principle suggests that the low energy effective field theory of gravity, as used to describe perturbative quantum fields about some background has far too many states. It is then natural that any quantum error correcting code with such a quantum field theory as the code subspace is not isometric. We discuss how this framework can naturally arise in an algebraic QFT treatment of a family of CFT with a large-$N$ limit described by the single trace sector. We show that an isometric code can be recovered in the $N \rightarrow \infty$ limit when acting on fixed states in the code Hilbert space. Asymptotically isometric codes come equipped with the notion of simple operators and nets of causal wedges. While the causal wedges are additive, they need not satisfy Haag duality, thus leading to the possibility of non-trivial entanglement wedge reconstructions. Codes with complementary recovery are defined as having extensions to Haag dual nets, where entanglement wedges are well defined for all causal boundary regions. We prove an asymptotic version of the information disturbance trade-off theorem and use this to show that boundary theory causality is maintained by net extensions. We give a characterization of the existence of an entanglement wedge extension via the asymptotic equality of bulk and boundary relative entropy or modular flow. While these codes are asymptotically exact, at fixed $N$ they can have large errors on states that do not survive the large-$N$ limit. This allows us to fix well known issues that arise when modeling gravity as an exact codes, while maintaining the nice features expected of gravity, including, among other things, the emergence of non-trivial von Neumann algebras of various types.

Figures (7)

• Figure 1: (left) Some examples of causal regions on $S^{d-1} \times \mathbb{R}$. (right) The additivity property we use in this paper considers spacetime covers of $O$ by causal diamonds.
• Figure 2: The case of AdS/CFT. (left) The causal wedge is the spacetime region $J^+(O) \cap J^-(O)$. The algebra of fields will be associated to the causal completion of this regions, dubbed the causal domain. This is a slightly larger region as discussed in Appendix \ref{['app:causal']}. The picture here is a cartoon showing a cross section of a $D>2$ spacetime through the causal wedge. (right) A cartoon of the symplectic flux deformation argument is given in the main text. The bulk symplectic form is evaluated on a time slice $\Sigma$, but can be deformed to the boundary where it becomes a product of sources and response.
• Figure 3: In AdS/CFT the additivity property of the causal domain comes from a conjecture for the dynamics of general bulk QFT which is supported by Borchers timelike tube theorem. See Appendix \ref{['app:causal']}.
• Figure 4: The extension of the causal wedge to the entanglement wedge can give back Haag duality $\mathcal{E}(O') = \mathcal{E}(O)'$.
• Figure 5: (left) Cartoon of requirements on the size of the approximating type-I factor $q(N)$ as a function of $N$. If the function is chosen to lie on the gray line, then we can use the maps $\beta_{N,q(N)}$ to give asymptotic reconstruction maps. The function must lie at or below the gray curve since larger code subspaces are problematic at fixed $N$ due to holographic bounds. (right) Different possibilities for codes in holographic theories. Most discussions of QEC in AdS/CFT have centered around small codes. Large codes have also been considered more recently Hayden:2018khnAkers:2021fut, and involve signification modifications to the standard error correcting the story. We straddle the two worlds along the weak convergence line.

Theorems & Definitions (60)

• Definition 1
• Definition 2
• Definition 3: Single trace algebra
• Definition 4: Large-$N$ sector
• Definition 5
• Definition 6
• Definition 7
• Remark 1
• Lemma 1
• proof