Relative Entropy and Bekenstein-Type Bounds

Positivity of relative entropy yields a precise regional entropy–energy inequality whenever the reference modular Hamiltonian is known. The result compares one state with one reference on one algebra. It avoids a regulator-dependent absolute entropy, but it is not a universal bound involving only total energy and a guessed radius.

Required background. Relative entropy in QFT supplies the algebraic quantity, and conformal flow for a ball supplies the local vacuum modular Hamiltonian.

Helpful background. Passivity and work clarify why modular energy is not automatically extractable mechanical work.

Positivity as the bound

Restrict a state $\rho$ and reference $\sigma$ to the same region $\mathcal R$. When the differences are defined,

$$S(\rho\Vert\sigma) =\Delta\langle K_\sigma\rangle-\Delta S_{\mathcal R}\ge0,$$

where $K_\sigma=-\log\sigma$ in a regulated type-I description and the relative modular operator supplies the continuum definition. Therefore

$$\Delta S_{\mathcal R}\le\Delta\langle K_\sigma\rangle .$$

For the vacuum of a CFT reduced to a ball $B_R$ at $t=0$,

$$K_B=2\pi\int_{|\mathbf x|<R}d^{d-1}x\, \frac{R^2-|\mathbf x|^2}{2R}\,T_{00}(\mathbf x) +c\,\mathbf1 .$$

The constant cancels in expectation differences. Positivity now bounds the vacuum-subtracted entropy by a weighted energy integral. The weight vanishes at the entangling surface; replacing it by $R$ gives a weaker estimate only under an additional sign assumption on the energy density.

This formulation, emphasized by Casini 2008, §§ 2–3, resolves the leading vacuum-area divergence because $\Delta S$ compares states with the same local ultraviolet structure.

Localized excitation in a ball

Prepare a smooth finite-energy excitation supported well inside $B_R$. Compute the stress profile and

$$\Delta\langle K_B\rangle =2\pi\int_{B_R}d^{d-1}x\, \frac{R^2-r^2}{2R}\,\Delta\langle T_{00}\rangle .$$

Compute $\Delta S_B$ with the same regulator and take the continuum difference. The residual

$$S(\rho_B\Vert\sigma_B) =\Delta\langle K_B\rangle-\Delta S_B$$

must be nonnegative. For an infinitesimal state perturbation, the first law gives equality at first order; positivity begins at quadratic order. A finite excitation need not saturate.

The relative-entropy branch requires a region, a reference state, and its modular Hamiltonian. Its weighted modular energy should not be confused with a timelike or null stress-tensor average. The diagram is schematic.

Boundaries of the inference

Move the excitation toward $\partial B_R$ and the modular weight decreases. A claim based on unweighted total energy can then fail even though the exact relative-entropy inequality remains true. Changing the reference from vacuum to a thermal state changes both $K_\sigma$ and the subtraction. Changing the region changes the algebra and cannot be treated as a harmless relabeling.

Unsubtracted entropy, a changed reference, or a replaced modular weight invalidates the regional bound. Positivity protects the exact relative-entropy expression, not every heuristic $ER$ formula. The map is schematic.

Exercises

Show that $0\le(R^2-r^2)/(2R)\le R/2$ inside the ball and derive a bound on $\Delta\langle K_B\rangle$ when $\Delta\langle T_{00}\rangle\ge0$ pointwise.

Solution

The weight is maximal at $r=0$ and vanishes at $r=R$. Thus $\Delta\langle K_B\rangle\le\pi R\,\Delta E_B$, and positivity gives $\Delta S_B\le\pi R\,\Delta E_B$ under the stated pointwise sign assumption.

References

• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25 (2008): 205021. DOI.