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Energy-Bounded Reconstruction and the Selective Entanglement Wedge

Brandon Christopher Russell

Published Mar 26, 2026• 1 citation
Publisher

Abstract

Version 5 (October 2026), interim correction: an independent review found that the causal-to-entanglement-wedge interpolation fails on the paper's own spherical example, that the Margolus–Levitin step bounds the modular parameter from below rather than above, and that the Python's-lunch exponent is not derived. These results should not be cited until the revised version is posted; the algebraic construction and the knapsack structure are unaffected. Details are in the Revision History of the PDF.We construct a resource-constrained framework for holographic bulk reconstruction by combining three existing results: the Chandrasekaran-Penington-Witten (CPW) Type II∞ algebra for boundary subregions, the JLMS decomposition of the modular Hamiltonian, and the Margolus-Levitin quantum speed limit applied to modular flow. Imposing a finite energy budget ⟨Kbdy⟩≤B on the boundary modular Hamiltonian simultaneously constrains the horizon area and bulk matter content via JLMS, and limits the rate of modular-flow-based reconstruction via the quantum speed limit. This produces a position-dependent reconstruction cost κ(x) that increases with bulk depth, inducing an effective metric dseff2=κ2(x)gμνdxμdxν. The accessible bulk region—the selective entanglement wedge—is determined by a continuous knapsack optimization that interpolates between the causal wedge (B→0) and the full entanglement wedge (B→∞).v3: removed inadvertent citation to QPI patent-track Zenodo deposit (citekey error in body cite of §9.4 routed to the QPI entry instead of the intended Rate-Distortion companion; corrected by adding the Rate-Distortion bibitem and repointing the cite); removed orphan "Virtualized Geometry" bibitem (subsumed under Spacetime as Optimal Allocation); updated three self-citation DOIs to concept DOIs; fixed body-text author attribution at §9.4(1) (Klinger, Kudler-Flam, and Satishchandran) and an equation subscript typo at Eq. 7. No content changes to the paper body.

DOI
10.5281/zenodo.19247166

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Version 5 (October 2026), interim correction: an independent review found that the causal-to-entanglement-wedge interpolation fails on the paper's own spherical example, that the Margolus–Levitin step bounds the modular parameter from below rather than above, and that the Python's-lunch exponent is not derived. These results should not be cited until the revised version is posted; the algebraic construction and the knapsack structure are unaffected. Details are in the Revision History of the PDF.We construct a resource-constrained framework for holographic bulk reconstruction by combining three existing results: the Chandrasekaran-Penington-Witten (CPW) Type II∞ algebra for boundary subregions, the JLMS decomposition of the modular Hamiltonian, and the Margolus-Levitin quantum speed limit applied to modular flow. Imposing a finite energy budget ⟨Kbdy⟩≤B on the boundary modular Hamiltonian simultaneously constrains the horizon area and bulk matter content via JLMS, and limits the rate of modular-flow-based reconstruction via the quantum speed limit. This produces a position-dependent reconstruction cost κ(x) that increases with bulk depth, inducing an effective metric dseff2=κ2(x)gμνdxμdxν. The accessible bulk region—the selective entanglement wedge—is determined by a continuous knapsack optimization that interpolates between the causal wedge (B→0) and the full entanglement wedge (B→∞).v3: removed inadvertent citation to QPI patent-track Zenodo deposit (citekey error in body cite of §9.4 routed to the QPI entry instead of the intended Rate-Distortion companion; corrected by adding the Rate-Distortion bibitem and repointing the cite); removed orphan "Virtualized Geometry" bibitem (subsumed under Spacetime as Optimal Allocation); updated three self-citation DOIs to concept DOIs; fixed body-text author attribution at §9.4(1) (Klinger, Kudler-Flam, and Satishchandran) and an equation subscript typo at Eq. 7. No content changes to the paper body.
In the context of quantum theories of spacetime, one overarching question is how quantum information in the bulk spacetime is encoded holographically in boundary degrees of freedom.It is particularly interesting to understand the correspondence between bulk subregions and boundary subregions in order to address the emergence of locality in the bulk quantum spacetime.For the AdS/CFT correspondence, it is known that this bulk information is encoded redundantly on the boundary in the form of an error-correcting code.Having access only to a subregion of the boundary is as if part of the holographic code has been damaged by noise and rendered inaccessible.In quantum-information science, the problem of recovering information from a damaged code is addressed by the theory of universal recovery channels.We apply and extend this theory to address the problem of relating bulk and boundary subregions in AdS/CFT, focusing on a conjecture known as entanglement wedge reconstruction.Existing work relies on the exact equivalence between bulk and boundary relative entropies, but these are only approximately equal in bulk effective field theory, and in similar situations it is known that predictions from exact entropic equalities can be qualitatively incorrect.We show that the framework of universal recovery channels provides a robust demonstration of the entanglement wedge reconstruction conjecture as well as new physical insights.Most notably, we find that a bulk operator acting in a given boundary region's entanglement wedge can be expressed as the response of the boundary region's modular Hamiltonian to a perturbation of the bulk state in the direction of the bulk operator.This formula can be interpreted as a noncommutative version of Bayes's rule that attempts to undo the noise induced by restricting to only a portion of the boundary.To reach these conclusions, we extend the theory of universal recovery channels to finite-dimensional operator algebras and demonstrate that recovery channels approximately preserve the multiplicative structure of the operator algebra.
Bulk reconstruction is a central problem in AdS/CFT, and entanglement wedge reconstruction is its subregion version. We argue that this subregion statement should be separated from the stronger holographic quantum error correction interpretation, in which one region-independent logical bulk operator has code-preserving representatives in several boundary regions. A simple locality argument shows that such a common reconstruction must commute with the code-preserving local algebras in the complementary regions. This is the mechanism realized in HaPPY-type codes: the erased regions are blind to a protected logical algebra. An ordinary finite NNN holographic CFT does not have such a protected invisible sector for supergravity fields. Its low-energy local observables, in particular, suitably smeared stress tensors, detect the physical support and gravitational dressing of ordinary bulk operators, up to possible center or superselection data. Thus, there is no such holographic quantum error correction and the N=∞N=\inftyN=∞ agreement of global and subregion HKLL formulae is a free-theory statement. What remains is entanglement wedge reconstruction without holographic quantum error correction, or subregion complementarity: each boundary region has its own code-preserving low-energy algebra and its own region-adapted bulk interpretation, rather than a shared logical operator.
Entanglement wedge reconstruction provides the sharpest formulation of bulk locality in AdS/CFT, identifying the entanglement wedge as the largest bulk region reconstructible from a boundary subregion. While this statement is well understood at leading order in the large N expansion through both modular flow and Petz map reconstruction, its extension beyond the semiclassical regime has remained unclear. In this work, we show that quantum error correction provides the natural framework for addressing this problem. At leading order, we reproduce the known leading-order reconstruction formulas, and by employing the twirled Petz map, we obtain a systematic and explicit method for incorporating subleading 1/N corrections. Beyond the leading approximation, we have obtained a bi-local correction controlled by a subleading Petz kernel. Our results establish a controlled approach to bulk reconstruction beyond large N and sharpening the role of quantum error correction in holography. This framework provides a controlled path toward understanding bulk reconstruction in fully quantum gravitational settings.
A bstract The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion A‾ \overline{A} A A ¯ , operators acting on the bulk qudit can be reconstructed as CFT operators on A‾ \overline{A} A A ¯ . This naturally fits within the framework of quantum error correction, with the CFT states containing the bulk qudit forming a code protected against the erasure of the boundary subregion A . In this paper, we set up and study a framework for relational bulk reconstruction in holography: given two code subspaces both protected against erasure of the boundary region A , the goal is to relate the operator reconstructions between the two spaces. To accomplish this, we assume that the two code subspaces are smoothly connected by a one-parameter family of codes all protected against the erasure of A , and that the maximally-entangled states on these codes are all full-rank. We argue that such code subspaces can naturally be constructed in holography in a “measurement-based” setting. In this setting, we derive a flow equation for the operator reconstruction of a fixed code subspace operator using modular theory which can, in principle, be integrated to relate the reconstructed operators all along the flow. We observe a striking resemblance between our formulas for relational bulk reconstruction and the infinite-time limit of Connes cocycle flow, and take some steps towards making this connection more rigorous. We also provide alternative derivations of our reconstruction formulas in terms of a canonical reconstruction map we call the modular reflection operator.
At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedure over bulk and boundary modular time, and hence it can be somewhat intractable to evaluate in practice. We show that a much simpler channel, the Petz map, is sufficient for entanglement wedge reconstruction for any code space of fixed finite dimension – no twirling is required. Moreover, the error in the reconstruction will always be non-perturbatively small. From a quantum information perspective, we prove a general theorem extending the use of the Petz map as a general-purpose recovery channel to subsystem and operator algebra quantum error correction.
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→0G \to 0G→0. As a result, precise arguments controlling error terms have focused on what we call `small' codes which, as G→0G \to 0G→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→0G \to 0G→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.