Modular Response, Relative Entropy, and Emergent-Gravity Claims

Relative entropy and modular response connect quantum information to semiclassical gravity with remarkable precision—but only through a declared holographic dictionary, operator algebra, region, state family, and perturbative order. The equality of boundary and bulk relative entropy established by Jafferis, Lewkowycz, Maldacena, and Suh 2016 is the central controlled example. This chapter develops the strongest relations from modular reconstruction to canonical energy and QNEC while making every gravitational inference and nonconverse explicit.

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.

Enter this chapter

Two routes share the same checkpoints.

• Modular route. Fix the algebra and reference state, match boundary and bulk relative entropy, use modular flow for wedge reconstruction, then study shape response, information geometry, crossed products, and Berry transport.
• Gravity-inference route. Import the boundary first law and holographic entropy dictionary, derive the all-ball integral identity, invert it at linear order, match second-order relative entropy to canonical energy, and stop before nonlinear claims unless the additional interaction data are present.

Every route must state whether an equality is exact in QFT, leading in $G_N$, perturbative in a state amplitude, conditional on a code subspace, or conjectural.

Chapter map

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.

The central implication chain

For a boundary ball $B$ and nearby state,

$$\delta S_B=\delta\langle K_B\rangle$$

is an exact information identity. The gravitational step adds

$$\delta S_B \xleftrightarrow{\ \mathrm{RT}\ } \frac{\delta A(\chi_B)}{4G_N}, \qquad \delta\langle T_{\mu\nu}\rangle \xleftrightarrow{\ \mathrm{dictionary}\ } h_{\mu\nu}^{(d)}.$$

A covariant phase-space identity then expresses the difference as a weighted integral of the linearized bulk equation. Equality for all balls and an injective transform yield the local linearized equation. RT, the stress-tensor map, locality, gauge control, and inversion are inputs; none follows from the first law alone.

At second order,

$$S(\rho_B(\lambda)\Vert\rho_B(0)) =\frac{\lambda^2}{2}\mathcal E_B(h,h)+O(\lambda^3),$$

with normalization conventions understood. Positivity constrains the canonical energy of the matched perturbation. Cubic interactions are new data and are not fixed by this quadratic form.

Objects that must remain distinct

Object	Domain	Strongest routine conclusion	Unsupported promotion
Entanglement first law	any smooth faithful state family	equality of first variations	Einstein dynamics without a dictionary
JLMS relation	matched code-subspace algebras and perturbative order	boundary/bulk modular and relative-entropy match	exact all-state finite-N identity
Canonical energy	linearized physical perturbations with surface terms	quadratic stability constraint	nonlinear global stability
Crossed-product entropy	declared algebra, weight, area mode, and regime	generalized-entropy realization in the model	unique algebra of every gravitational region
QNEC	theorem’s QFT, state, geometry, and renormalization domain	local null-energy/entropy inequality	general quantum focusing conjecture
Modular Berry curvature	smooth modular family and zero-mode group	covariant transport; bulk match in symmetric examples	unique geometry from one holonomy

Checks before using an implication

1. Are boundary and bulk algebras, including centers, actually matched?
2. Is the expansion order in state amplitude separate from the order in $G_N$ or $1/N$?
3. Does the extremal or quantum extremal surface remain on one branch?
4. Are gauge transformations trivial at all relevant boundaries, or do they carry charges?
5. Are symplectic, entropy, and stress-tensor counterterms in the same scheme?
6. Is the family of regions or states rich enough for the claimed inversion?
7. Has the reverse implication been proved, or merely assumed?

These questions turn a slogan about emergence into a reproducible statement.

Scientific boundaries

Quantum Information in QFT owns the definitions and general theorems of relative entropy, modular theory, information metrics, and energy inequalities. QFT in Curved Spacetime owns semiclassical equations, generalized entropy, focusing, and QFC status. This chapter owns the holographic matching and the chiefly linearized gravitational inferences.

Current algebraic proposals continue to develop. The dated summary page records primary results through 10 August 2026; it does not convert a recent preprint into consensus or an exact nonperturbative definition.

Review the chapter

1. Why does the area term cancel from relative entropy but remain in the modular-operator relation?
2. Why are all boundary balls needed to infer a local linearized equation?
3. How does modular evolution turn an antisymmetric symplectic form into a positive information metric?
4. Which choices determine a crossed-product entropy?
5. Why can QNEC be proved while the general quantum focusing statement remains conjectural?
6. Give two bulk actions with identical quadratic data but different interactions; which “emergence” claim fails?

A complete answer identifies state subtraction for question 1, transform injectivity for question 2, canonical energy for question 3, algebra/weight/area/trace data for question 4, distinct domains and logical directions for question 5, and the insufficiency of first- and second-order information for question 6.

Where to go next

Entanglement Wedges and Holographic Quantum Error Correction develops recovery maps, complementary reconstruction, centers, and finite-N errors. Holographic Complexity Proposals and Diagnostics treats inequivalent geometric conjectures whose status is weaker than the relative-entropy identities here.

Chapter-scale structure and validity checks

The chapter-scale structure map locates this page’s result inside the full reasoning chain. Follow the solid arrows through the declared inputs and checks; the dashed final arrow marks the point where an additional inference would be required.

Information identities constrain gravity only after the dictionary, code sector, semiclassical expansion, and boundary terms are fixed. The diagram is an original schematic, is not to scale, and uses the dashed final arrow to mark the claim boundary.

Accessible figure data (JSON)

The companion validity map turns three common overclaims into explicit failure tests. Read each row from its declared object to the diagnostic, then compare the licensed conclusion with the dashed “not” endpoint.

Information identities constrain gravity only after the dictionary, code sector, semiclassical expansion, and boundary terms are fixed. Each row pairs a diagnostic with the strongest supported conclusion and an explicitly unsupported promotion. The diagram is an original schematic and is not to scale.

Accessible figure data (JSON)

Claim-domain comparison

The table below gives a screen-reader-friendly comparison of three representative claims. It keeps the required declaration, approximation status, evidence timing, counterevidence, falsifier, failure condition, and licensed conclusion in one reading order.

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.	linear-response identity	nonlinear field equations	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.	second-order positivity check	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.	constraint and boundary-term check	a unique quantum-gravity action	conditional linearized equations

Download the structured table data (JSON).

References

• Faulkner, Thomas, Monica Guica, Thomas Hartman, Robert C. Myers, and Mark Van Raamsdonk. “Gravitation from Entanglement in Holographic CFTs.” Journal of High Energy Physics 2014, 051 (2014). DOI; arXiv:1312.7856.
• 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.
• Lashkari, Nima, and Mark Van Raamsdonk. “Canonical Energy Is Quantum Fisher Information.” Journal of High Energy Physics 2016, 153 (2016). DOI; arXiv:1508.00897.