The JLMS formula in a large code with approximate error correction
Published 1 Jan 2026 in hep-th and quant-ph | (2601.00442v1)
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 singlelarge' 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.
The paper presents a large code framework that extends the JLMS formula to include nonperturbative bulk excitations through approximate error correction.
It systematically controls error propagation using log-stability conditions and small code constructions that form a direct sum over RT/HRT area windows.
The method ensures robust modular flow and bulk reconstruction by managing subsystem orthogonality and sector mixing in a holographic setting.
Summary of "The JLMS formula in a large code with approximate error correction" (2601.00442)
Introduction and Background
The JLMS formula, originally derived by Jafferis, Lewkowycz, Maldacena, and Suh, rigorously expresses the relationship between modular Hamiltonians in the context of AdS/CFT and underpins the operator-algebra quantum error correction (OAQEC) perspective on holography. Standard treatments demonstrate that for a code subspace describing perturbative excitations around a fixed bulk geometry, the code realizes exact error correction with complementary recovery in the G→0 limit (where G is Newton’s constant). Explicitly, the projected JLMS formula relates the CFT modular Hamiltonian in region R to an area term and a bulk modular Hamiltonian in the entanglement wedge, restricted by a projector to the code subspace.
However, this structure is known to break down in the presence of order-one bulk excitations, and complications arise when considering code subspaces containing superpositions of distinct semiclassical backgrounds. Perturbative expansions are insufficient to capture the modular flow generated by bulk states differing at O(1), and attempts to naively enlarge the code encounter pathologies due to highly degenerate density matrices in the G→0 limit, leading to ill-defined modular Hamiltonians.
Main Contributions
This work constructs an explicit framework for implementing the JLMS formula and its exponentiated (modular flow) variant in "large" codes built systematically from "small" codes, each satisfying approximate OAQEC properties. These large codes can accommodate states superposing over distinct semiclassical geometries, thus capturing nonperturbative excitations and enabling modular flow that shifts the background geometry by order-one amounts. The code is constructed as a direct sum over blocks labeled by small windows of RT/HRT areas.
Careful analysis delineates the error terms in the JLMS relation for this structure. For each small code–subspace, the quantum-corrected HRT (Faulkner-Lewkowycz-Maldacena, FLM) formula is assumed to hold up to a small error, and conditions are established to control cumulative errors in the large code, even when considering states combining support across sectors. The framework distinguishes between perturbatively small error terms (ϵ) and non-perturbatively small ones (ε).
Several technical obstacles are addressed:
Degeneracy of modular Hamiltonians: Since states may have exponentially small eigenvalues in $1/G$ in the G→0 limit, error terms may be dramatically enhanced. The authors introduce a log-stability condition (spectral gap condition on density matrices) to render all variations and errors finite and controllable.
Error propagation: The connection between small and large code error bounds is carefully established, showing that if FLM and code structure are controlled with perturbative and nonperturbative accuracy in the small codes, validity of JLMS and modular flow in the large code follows for suitably restricted classes of states.
Subsystem orthogonality: The construction and estimates employ a refined notion of approximate subsystem orthogonality in the boundary image of the code, which ensures that off-diagonal blocks are suppressed and modular Hamiltonians decompose appropriately.
A systematic hierarchy of JLMS-like relations is established:
On small codes, approximate projected JLMS and exponentiated JLMS relations hold for all log-stable states, with error bounds polynomially controlled by FLM and isometry parameters.
On the large code, analogous (block-diagonalized) relations hold for either:
States whose support is confined (aligned) within well-populated sectors (making error terms negligible),
Or for all states, provided the code-weights are sufficiently "smooth" functions (akin to a Gaussian envelope suppressing problematic sector mixing).
Technical Results
The main technical results can be grouped as:
Projected JLMS Relation:
For any log-stable state in the small code, the projected JLMS formula holds up to an error
proportional to ϵ/δ​+ϵ′, where ϵ quantifies FLM inaccuracy and δ is the log-stability lower bound on eigenvalues.
Exponentiated JLMS Relation: For (approximately) flat entanglement spectra, the exponentiated JLMS relation holds (i.e., relating modular evolutions on bulk and boundary) with errors scaling as a function of code parameters and R\'enyi-FLM inaccuracy for complex replica indices.
Large Code Construction: The "large" code is assembled as a direct sum of "small" codes labeled by area windows. The modular Hamiltonian and modular flow on this code (including states that superpose over area windows, i.e., distinct classical backgrounds) are shown to satisfy block-extended JLMS relations with error bounds precisely characterized.
Subsystem Orthogonality and Alignment/Smoothness:
Two regimes are studied:
Aligned states: The modular flow is controlled for states whose support is concentrated within the same sectors as the reference state.
Smoothness: If the distribution over area sectors varies slowly, the JLMS formula holds in operator norm for arbitrary states.
Practical AdS/CFT Context: The formulas and error bounds are shown to be applicable for codes in AdS/CFT built by truncating perturbative bulk Hilbert spaces by energy and area windows, and sector orthogonality is well-justified by the physics of different RT/HRT sectors.
Implications and Discussion
The principal upshot is a rigorous prescription for using the JLMS formula, and thus bulk reconstruction and modular flow techniques, in large codes that accommodate finite G effects and sector superpositions in the bulk. This brings the structure of the holographic code closer to the expectations in full quantum gravity, theoretically permitting analyses of states such as those describing black hole microstates, superpositions of geometries, and time-dependent dynamics relevant for the black hole information problem.
Technically, the work elucidates how the sector structure of the code is interlaced with spectral properties and modular theory, and connects to the recent developments in type II von Neumann algebras in gravity [e.g., "Generalized entropy for general subregions in quantum gravity" (Jensen et al., 2023), "An algebra of observables for de Sitter space" (Chandrasekaran et al., 2022)].
A key requirement throughout is the log-stability criterion, which forces a cut-off on the dimension of each block. While this is not a conceptual issue for effective field theory applications (finite code dimension per area window), the physical interpretation of exponential density-of-states growth near the semi-classical limit becomes relevant. Extensions to relax this requirement may be amenable using von Neumann algebraic methods.
The presented framework systematizes error accounting in JLMS and modular reconstruction, and provides analytic and computational control for explicit error bounds in AdS/CFT. These techniques are essential for future advances targeting real-time dynamics, operator reconstruction under large modular flow, and rigorous analyses of quantum gravity entropy formulae.
Conclusion
This work resolves a long-standing technical obstacle in modular Hamiltonian and bulk reconstruction theory by constructing a large code, built from controlled approximate small codes, in which both projected and exponentiated JLMS formulas hold with quantitatively controlled errors for an appropriate class of states. The "large code" structure allows for non-perturbative area fluctuations and sector superpositions, as required by true semiclassical gravitational states, while preserving the modular Hamiltonian correspondence and error correction properties foundational to holography. This construction is applicable in the AdS/CFT context for any code with finite sector dimensions and controllable FLM accuracy, and the methods developed here have implications for rigorous quantum-gravity entropy, real-time dynamics, and modular flow beyond leading order.
