The JLMS formula in a large code with approximate error correction

TL;DR

The paper addresses the limitation that exact JLMS with complementary recovery is only precise as $G\to 0$ in small code subspaces, and constructs a single large code built from many small codes to accommodate modular flows between distinct classical backgrounds. It provides explicit error bounds for both projected and exponentiated JLMS relations on the small code and large-code constructions, using a framework of FLM-based approximate entropies, log-stability, and approximate subsystem orthogonality. The main contributions are (i) a detailed error-analysis strategy that splits sources into FLM, tail, and alignment terms, (ii) rigorous operator-norm JLMS and Rényi-flavored JLMS bounds on small codes, and (iii) a large-code assembly with aligned/exponentiated JLMS results and corresponding operator-norm bounds, under smoothness and alignment assumptions. The results offer a principled route to robust bulk reconstruction and modular-flow analysis in AdS/CFT at finite $G$, with potential links to emergent type II algebras and the ModFlow program for semiclassical bulk dynamics.

Abstract

Gauge/gravity duality is often described as a quantum error correcting code. However, as seen in the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula, exact quantum error correction with complementary recovery (and thus entanglement wedge reconstruction) emerges only in the limit $G \to 0$. As a result, precise arguments controlling error terms have focused on what we call `small' codes which, as $G \to 0$, describe only perturbative excitations near a given classical solution. Such settings are quite restrictive and, in particular, they prohibit discussion of any modular flow that would change the classical background. As a result, they forbid consideration of modular flows generated by semiclassical bulk states at order-one modular parameters. In contrast, we present a single `large' code for the bulk theory that can accommodate such flows and, in particular, in the $G \to 0$ limit includes superpositions of states associated with distinct classical backgrounds. This large code is assembled from small codes that each satisfy an approximate Faulkner-Lewkowycz-Maldacena formula. In this extended setting we clarify the meaning of the (approximate) JLMS relation between bulk and boundary modular Hamiltonians and quantify its validity in an appropriate class of states.

Figures (3)

• Figure 1: The QEC we consider is a network made with random tensor $T$ with the $B$ leg projected onto a state $\ket{\phi}$ with non-maximal entanglement.
• Figure 2: The approximate description of the spectrum of $\rho_A$ in the regime of interest. The important feature is that there are no zero eigenvalues.
• Figure 3: The saddle $g$ that computes $\operatorname{Tr}_R\left( \tilde{\sigma}^\alpha_R (\tilde{\rho}^\alpha_R)^{n-2} \tilde{\sigma}^\alpha_R\right)$. It is constructed by gluing together copies of saddles $g_1$ (yellow) and $g_2$ (light blue) around the HRT surface in a cyclic manner.

Theorems & Definitions (41)

• Definition 1: Small code
• Definition 2: Approximate FLM
• Definition 3: Approximate isometry of the small code
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof
• theorem 1: Approximate projected JLMS in a small code
• proof