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arXiv:2609.14165

Modular flow and the area operator

Xi Dong, Donald M. Marolf, Pratik Rath

Published Sep 12, 2026• 0 citations
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Abstract

In the context of the AdS/CFT correspondence, the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula relates the boundary modular Hamiltonian to the HRT area operator and the corresponding bulk modular Hamiltonian. In standard discussions of a holographic code built from perturbative excitations on a fixed classical background, the $A/4G$ term dominates over the bulk modular Hamiltonian in the limit $G \to 0$. However, this setup is insufficient to describe the action of boundary modular flows that change the classical background. In contrast, our recent discussion of `large' codes provides a useful framework to obtain a JLMS formula that relates bulk and boundary modular flows acting on appropriate states in the code. Here, one sees that the bulk modular Hamiltonian is not in fact subdominant when the modular flow defined by one state acts on another state peaked around a sufficiently different classical background. Instead, the bulk modular Hamiltonian can be of order $1/G$ and can dominate over the $A/4G$ term. Nevertheless, we show that if the two states have the same classical background except for data conjugate to the HRT area then, under mild assumptions and over a large range of flow parameters, the modular-flowed state is described by a classical background given by the HRT-area flow alone.

DOI
10.48550/arxiv.2609.14165

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In the context of the AdS/CFT correspondence, the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula relates the boundary modular Hamiltonian to the HRT area operator and the corresponding bulk modular Hamiltonian. In standard discussions of a holographic code built from perturbative excitations on a fixed classical background, the A/4GA/4GA/4G term dominates over the bulk modular Hamiltonian in the limit G→0G \to 0G→0. However, this setup is insufficient to describe the action of boundary modular flows that change the classical background. In contrast, our recent discussion of `large' codes provides a useful framework to obtain a JLMS formula that relates bulk and boundary modular flows acting on appropriate states in the code. Here, one sees that the bulk modular Hamiltonian is not in fact subdominant when the modular flow defined by one state acts on another state peaked around a sufficiently different classical background. Instead, the bulk modular Hamiltonian can be of order 1/G1/G1/G and can dominate over the A/4GA/4GA/4G term. Nevertheless, we show that if the two states have the same classical background except for data conjugate to the HRT area then, under mild assumptions and over a large range of flow parameters, the modular-flowed state is described by a classical background given by the HRT-area flow alone.
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.
Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
Abstract Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
In the context of the holographic duality, the entanglement entropy of ordinary QFT in a subregion in the boundary is given by a quarter of the area of an minimal surface embedded in the bulk spacetime. This rule has been also extended to a suitable one-parameter generalization of the von-Neuman entropy hatSn\\hat{S}_nhatSn​ that is related to the Renyi entropies SnS_nSn​, as given by the area of a \\emph{cosmic brane} minimally coupled with gravity, with a tension related to nnn that vanishes as nto1n\\to1nto1, and moreover, this parameter can be analytically extended to arbitrary real values. However, the brane action plays no role in the duality and cannot be considered a part of the theory of gravity, thus it is used as an auxiliary tool to find the correct background geometry. In this work we study the construction of the gravitational (reduced) density matrix from holographic states, whose wave-functionals are described as euclidean path integrals with arbitrary conditions on the asymptotic boundaries, and argue that in general, a non-trivial Hayward term must be haven into account. So we propose that the gravity model with a coupled Nambu-Goto action is not an artificial tool to account for the Renyi entropies, but it is present in the own gravity action through a Hayward term. As a result we show that the computations using replicas simplify considerably and we recover the holographic prescriptions for the measures of entanglement entropy; in particular, derive an area law for the original Renyi entropies (SnS_nSn​) related to a minimal surface in the nnn replicated spacetime. Moreover, we show that the gravitational modular flow contains the area operator and can explain the Jafferis-Lewkowycz-Maldacena-Suh proposal.
In this work we study the Tomita-Takesaki construction for a family of excited states that, in a strongly coupled CFT---at large NNN---correspond to coherent states in an asymptotically AdS spacetime geometry. We compute the modular flow and modular Hamiltonian associated to these excited states in the Rindler wedge and for a ball shaped entangling surface. Using holography, one can compute the bulk modular flow and construct the Tomita-Takesaki theory for these cases. We also discuss generalizations of the entanglement regions in the bulk and how to evaluate the modular Hamiltonian in a large N approximation. Finally, we extend the holographic Banks, Douglas, Horowitz and Matinec (BDHM) formula to compute the modular evolution of operators in the corresponding CFT algebra, and propose this as a more general prescription.
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.