Boundary Relative Entropy and Bulk Modular Data

Boundary relative entropy can equal bulk relative entropy in a semiclassical code sector even though boundary and bulk entropies separately contain geometric terms. The cancellation is precise only after the boundary region, bulk wedge, operator algebras, reference state, renormalization, and order in $G_N$ are fixed. It is not an exact finite-N identity for arbitrary states.

Required background. Relative Entropy for QFT States supplies the information quantity. FLM Corrections and Holographic Use of Imported Generalized Entropy and QES supplies the area-plus-bulk-entropy expansion.

Helpful background. Araki Relative Entropy and Regulated Limits gives the algebraic definition; Relative Entropy and Modular Horizon Laws gives the gravitational comparison; Relative Entropy and Bekenstein-Type Bounds supplies the localization inequality.

First application. Evaluate relative entropy for nearby states in a ball-shaped region and compare boundary and bulk modular contributions at leading order.

Modular Hamiltonians in a code sector

For faithful states $\rho_A$ and $\sigma_A$ on a regulated boundary region,

$$S(\rho_A\Vert\sigma_A) =\operatorname{Tr}\rho_A(\log\rho_A-\log\sigma_A) =\Delta\langle K_A^\sigma\rangle-\Delta S_A,$$

where $K_A^\sigma=-\log\sigma_A$ and each $\Delta$ subtracts the reference state. In continuum QFT, the algebraic relative entropy is primary; the two terms on the right can be separately divergent even when their difference is finite.

In a semiclassical holographic code subspace, the JLMS relation has the schematic operator form

$$K_A^{\mathrm{CFT}} =\frac{\widehat{\mathcal A}_\chi}{4G_N} +K_a^{\mathrm{bulk}}+O(G_N),$$

on code states. $a$ is the entanglement wedge and $\widehat{\mathcal A}_\chi$ is an area operator or center-valued geometric term associated with the extremal surface. The corresponding entropy expansion is

$$S_A^{\mathrm{CFT}} =\frac{\langle\widehat{\mathcal A}_\chi\rangle}{4G_N} +S_a^{\mathrm{bulk}}+O(G_N).$$

Subtracting the reference state makes the area contributions cancel, giving

$$S(\rho_A\Vert\sigma_A) =S(\rho_a\Vert\sigma_a)+O(G_N)$$

at the stated perturbative order Jafferis, Lewkowycz, Maldacena, and Suh 2016, §§2–4.

Nearby ball states

Let $A$ be a ball in the vacuum CFT and $\rho(\lambda)=\sigma+\lambda\delta\rho+O(\lambda^2)$. Relative entropy starts at quadratic order,

$$S(\rho(\lambda)\Vert\sigma) =\frac{\lambda^2}{2}\,\mathcal I_\sigma(\delta\rho,\delta\rho) +O(\lambda^3),$$

where $\mathcal I_\sigma$ is the appropriate quantum Fisher form. The leading $O(\lambda)$ equality is the entanglement first law; the nontrivial comparison is at $O(\lambda^2)$. In the bulk it becomes a positive relative-entropy or canonical-energy expression after gauge and surface terms are fixed.

This application has two expansions: the state amplitude $\lambda$ and the gravitational loop parameter. “Second order” must say which expansion is meant. A calculation quadratic in $\lambda$ can still be only leading or next-to-leading in $G_N$.

Algebra, center, and wedge choice

Gauge constraints and gravity can give a subregion algebra a center. The area term is then not an ordinary operator assigned independently to both complementary tensor factors. One must state the von Neumann algebra, its center, and whether relative entropy is computed algebraically or in an extended-factorization regulator.

Wedge phase transitions add another qualification. If $\rho$ and $\sigma$ select different quantum extremal surfaces, a single perturbative wedge algebra may not cover the interpolation. The relative-entropy equality is most controlled within a fixed code sector and fixed surface branch.

Adversarial controls

Drop the area term from the modular operator. Boundary and bulk entropy differences then fail to match at order $1/G_N$. The eventual cancellation in relative entropy does not license omitting the term before forming the difference.

Change the algebra. Adding edge or center observables changes the state restriction and relative entropy. An unchanged numerical formula cannot be transported between algebras without a map.

Cross a surface transition. Perturbation theory about one wedge can miss a competing QES. A discontinuous wedge choice invalidates the fixed-branch expansion even if each saddle is individually smooth.

Evidence ceiling

The leading JLMS relation establishes equality of boundary and bulk relative entropy for matched algebras and states in a semiclassical code subspace, with a geometric term in the modular operator and the stated corrections. It does not prove exact finite-N entanglement-wedge reconstruction, a global Hilbert-space factorization, or equality for states that leave the code sector.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Faulkner, Thomas, Aitor Lewkowycz, and Juan Maldacena. “Quantum Corrections to Holographic Entanglement Entropy.” Journal of High Energy Physics 2013, 074 (2013). DOI; arXiv:1307.2892.
• 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.