The JLMS formula in a large code with approximate error correction

TL;DR

This paper introduces a large quantum error correcting code in holography that can handle superpositions of classical backgrounds and approximate JLMS relations beyond the small code limit, advancing understanding of entanglement wedge reconstruction.

Contribution

It constructs a large code from small codes satisfying the Faulkner-Lewkowycz-Maldacena formula, enabling analysis of modular flows involving superpositions of classical states.

Findings

Large code accommodates superpositions of classical backgrounds.

Clarifies the approximate JLMS relation in extended settings.

Quantifies the validity of the JLMS formula for a broader class of states.

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.