Modular Response, Relative Entropy, and Emergent-Gravity Claims

Boundary relative entropy and modular response can constrain bulk dynamics within a specified holographic dictionary. The useful connections depend on the state family, region, code sector, and perturbative order.

Enter this chapter

Begin with Boundary Relative Entropy and Bulk Modular Data to compare boundary and bulk modular data in a controlled code sector.

Helpful background. Relative Entropy for QFT States, First Law of Entanglement and Quadratic Corrections, and Quantum Fisher Information in QFT supply the information-theoretic objects. QNEC on Curved Backgrounds supplies the theorem domains. FLM Corrections and Holographic Use of Imported Generalized Entropy and QES supplies the semiclassical entropy expansion.

Topics

1. Boundary Relative Entropy and Bulk Modular Data — Derives the area cancellation and leading JLMS relative-entropy equality.
2. Modular Flow Reconstruction and Bulk Modular Evolution — Represents wedge operators using modular frequencies and a controlled correlator inverse.
3. Entanglement First Law and Conditional Linearized Field-Equation Inference — Derives the all-ball integral identity and states the inversion hypotheses.
4. Canonical Energy and Second-Order Relative Entropy — Matches the information Hessian to the renormalized gravitational symplectic form.
5. Shape Deformations and Displacement Response — Connects modular shape variation to the extremal-surface Jacobi equation.
6. Quantum Fisher Information and Bulk Symplectic Forms — Distinguishes the symmetric information metric from the antisymmetric bulk form and identifies the modular complex structure.
7. Crossed-Product Gravitational Algebras and Generalized-Entropy Terms — Explains Type-III, Type-II, area-mode, and trace choices.
8. Modular Berry Transport and Bulk Connections — Develops zero-mode gauge freedom, curvature, and the symmetric-example bulk map.
9. Holographic ANEC, QNEC, and Focusing Arguments — Preserves theorem domains while applying energy inequalities to extremal surfaces and causality.
10. Beyond Linear Order: Conditional Emergent-Equation Claims — Identifies the higher correlators, entropy corrections, locality, and closure data missing from first-law arguments.
11. Emergent-Gravity Claims: Assumptions, Nonconverses, and Status — Compares the durable implications and records the literature boundary through 10 August 2026.

How the ideas fit together

The semiclassical relation between boundary and bulk modular data includes a geometric area contribution; in the corresponding relative-entropy comparison, the matched geometric terms cancel Jafferis, Lewkowycz, Maldacena, and Suh 2016, §§ 2–3. This connects a boundary information quantity to bulk state distinguishability under the assumed dictionary.

First-order entropy variations, second-order canonical energy, and shape deformations constrain different aspects of the bulk. They do not independently prove the dictionary or a complete nonlinear gravitational theory. The information metric and the observable algebra must remain fixed before a response coefficient is interpreted geometrically.

The comparison below separates the assumptions and tests needed for these different conclusions.

Claim-domain comparison

Representative claim domains and validity boundaries for Modular Response, Relative Entropy, and Emergent-Gravity Claims

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
entanglement first law	Declare reference state and modular operator; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary relative entropy → modular flow and first law → bulk canonical energy → positivity and shape checks → conditional gravity inference. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “linear-response identity” check is counterevidence to the promoted claim.	For a smooth state variation in the stated regulated domain, a first-order residual remains between entropy variation and reference-modular-energy variation.	Inferring nonlinear field equations from a first-order entropy relation.	first-order entropy relation
relative-entropy equality	Declare boundary and bulk algebras plus area term; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary relative entropy → modular flow and first law → bulk canonical energy → positivity and shape checks → conditional gravity inference. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “second-order positivity check” check is counterevidence to the promoted claim.	With algebras, reference states, and area terms matched, the proposed equality fails at the retained order or contradicts the required relative-entropy positivity.	Promoting a code-sector relation to exact equality for all states.	canonical-energy relation in a code sector
field-equation inference	Declare all balls or shapes and local dynamics assumptions; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary relative entropy → modular flow and first law → bulk canonical energy → positivity and shape checks → conditional gravity inference. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “constraint and boundary-term check” check is counterevidence to the promoted claim.	An omitted boundary or constraint term survives in the declared region family, invalidating the claimed conversion of the integral identity into local equations.	Inferring a unique quantum-gravity action from conditional linearized equations.	conditional linearized equations

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Where to go next

Continue to Entanglement Wedges and Holographic Quantum Error Correction to locate the bulk region associated with a specified boundary subregion.

References

• Jafferis, Daniel L., Aitor Lewkowycz, Juan Maldacena, and S. Josephine Suh. “Relative Entropy Equals Bulk Relative Entropy.” Journal of High Energy Physics 2016, 004 (2016). DOI; arXiv:1512.06431.