Relative Entropy and Bekenstein-Type Bounds

A precise Bekenstein-type inequality already exists in ordinary quantum field theory: compare a state with a reference state on the same spatial region, and the entropy difference cannot exceed the corresponding modular-energy difference. For admissible state pairs, the result is regulator-independent, operationally tied to distinguishability, and exact. It is not the universal gravitational slogan $S\leq 2\pi ER$, and its “energy” is generally a reference- and position-dependent modular quantity rather than the ordinary Hamiltonian.

Required background. Relative entropy in QFT supplies the continuum state-comparison quantity. Conformal flow for a ball supplies the exceptional case in which the vacuum modular Hamiltonian is a local stress-tensor integral.

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

For the chapter-wide orientation, use the entry map, the comparison of claim domains, and the controls that change conclusions.

Relative entropy gives a regional bound

Let $\mathcal A(B)$ be the observable algebra of one fixed region $B$. Compare two normal states on that same algebra: the state of interest $\rho_B$ and a faithful reference $\sigma_B$. In a common type-I regulator, define

$$D(\rho_B\Vert\sigma_B) =\operatorname{tr}\rho_B(\log\rho_B-\log\sigma_B), \qquad K_\sigma=-\log\sigma_B .$$

The support condition $\operatorname{supp}\rho_B\subseteq\operatorname{supp}\sigma_B$ makes the finite-dimensional expression finite; otherwise $D=+\infty$. Local QFT algebras need not possess density matrices, but Araki relative entropy supplies the regulator-independent algebraic object. Whenever the following differences are separately meaningful in a common scheme,

$$\Delta S_B=S(\rho_B)-S(\sigma_B), \qquad \Delta\langle K_\sigma\rangle =\langle K_\sigma\rangle_\rho-\langle K_\sigma\rangle_\sigma,$$

one has the identity

$$D(\rho_B\Vert\sigma_B) =\Delta\langle K_\sigma\rangle-\Delta S_B\geq0.$$

Thus

$$\boxed{\Delta S_B\leq\Delta\langle K_\sigma\rangle .}$$

This is just positivity of relative entropy, not an additional axiom about matter. It compares states rather than assigning a finite absolute entropy to a continuum region. Casini 2008, §4, Eqs. (17)–(19) gives this formulation and explains why its state subtraction avoids the naive species paradox.

For a differentiable family $\rho_B(s)$ with $\rho_B(0)=\sigma_B$, relative entropy begins at second order. Differentiating at $s=0$ therefore gives the entanglement first law

$$\left.\frac{dS_B}{ds}\right|_0 =\left.\frac{d\langle K_\sigma\rangle}{ds}\right|_0 .$$

Equality here is only a first-order statement. A finite displacement generally produces positive relative entropy and hence a strict gap; see Blanco et al. 2013, §2.

Why a CFT ball has a geometric modular energy

Work in $d$-dimensional Minkowski spacetime with the chapter’s metric convention. Let $B_R$ be the ball $|\mathbf x|<R$ on the $t=0$ slice, let $\sigma_B$ be the CFT vacuum restricted to $B_R$, and use the improved, renormalized stress tensor of the CFT. Conformal transport of Rindler modular flow gives

$$K_B =2\pi\int_{|\mathbf x|<R}d^{d-1}x\, \frac{R^2-|\mathbf x|^2}{2R}\,T_{00}(0,\mathbf x) +c_B\mathbf 1 .$$

The additive constant $c_B$ normalizes the state and cancels from expectation differences. The derivation and geometric hypotheses are explicit in Casini, Huerta, and Myers 2011, §2.1, Eq. (22). Positivity then gives

$$\Delta S_B \leq 2\pi\int_{B_R}d^{d-1}x\, \frac{R^2-r^2}{2R}\, \Delta\langle T_{00}(0,\mathbf x)\rangle .$$

The weight is largest at the center and vanishes at the entangling surface. Consequently, ordinary energy and modular energy contain different spatial information. If one additionally assumes $\Delta\langle T_{00}\rangle\geq0$ pointwise, then

$$0\leq \frac{R^2-r^2}{2R}\leq\frac R2 \quad\Longrightarrow\quad \Delta S_B\leq\Delta\langle K_B\rangle \leq\pi R\,\Delta E_B .$$

Without the pointwise sign assumption, the last step does not follow: a negative contribution multiplied by a smaller weight reverses the elementary comparison. Even with positive energy density, the coefficient here is $\pi R$ for a ball of radius $R$ under these definitions, not a license to import a different $2\pi ER$ convention.

Localized-excitation benchmark

Suppose a differentiable family of CFT states has, to first order in a dimensionless amplitude $\eta$, the positive energy profile

$$\Delta\langle T_{00}(x)\rangle =\eta E p(x), \qquad \int_{-R}^{R}p(x),dx=1.$$

Take $p$ to be supported inside the interval, with mean $x_0$ and variance $a^2$. At leading order the ordinary energy is $\Delta E_B=\eta E$, whereas the weighted integral is

$$\Delta\langle K_B\rangle =\frac{\pi\eta E}{R}\left(R^2-x_0^2-a^2\right)+O(\eta^2), \qquad \Delta S_B =\frac{\pi\eta E}{R}\left(R^2-x_0^2-a^2\right)+O(\eta^2).$$

The entropy equality at displayed order is the entanglement first law; the nonnegative relative entropy begins at $O(\eta^2)$. For $R=E=1$, $\eta=0.01$, and $a=0.1$, a centered packet gives $\Delta K_B=0.0311018$, while moving the same profile to $x_0=0.8$ gives $0.0109956$. In the narrow-packet limit, moving the center from $0$ to $0.9R$ reduces the result from $\pi\eta ER$ to $0.19\pi\eta ER$. This application assumes that the stated first-order stress profile comes from a valid state family; the moment calculation alone does not construct that state. It does show reproducibly that the same ordinary energy need not determine the exact regional bound.

Benchmark: a thermal state on a CFT interval

The first application can be reproduced without numerics. Take a unitary two-dimensional CFT of central charge $c$ on the infinite line. Let $B=(-R,R)$, use the vacuum as reference, and restrict a translation-invariant thermal state of inverse temperature $\beta$ to $B$. Assume the standard continuum interval entropy with the same short-distance regulator in the vacuum and thermal calculations. Define

$$z=\frac{2\pi R}{\beta}.$$

The thermal energy density and the vacuum modular Hamiltonian give

$$\Delta\langle T_{00}\rangle=\frac{\pi c}{6\beta^2}, \qquad \Delta\langle K_B\rangle =\frac{c z^2}{18}.$$

The interval entropy formula of Calabrese and Cardy 2004, §3 yields

$$\Delta S_B =\frac c3\log\!\left(\frac{\sinh z}{z}\right),$$

so the relative entropy is

$$D(\rho_{\beta,B}\Vert\rho_{0,B}) =c\left[ \frac{z^2}{18} -\frac13\log\!\left(\frac{\sinh z}{z}\right) \right].$$

For small $z$,

$$D=\frac{c z^4}{540}+O(z^6),$$

which exhibits first-law saturation at order $z^2$ and a positive distinguishability gap at the next order. A concrete check with $c=1$, $R=1$, and $\beta=10$ gives

$$z=0.62831853,\qquad \Delta K_B=0.02193245,\qquad \Delta S_B=0.02165086,$$

and hence $D=2.81593\times10^{-4}>0$. These numbers follow directly from the three displayed formulas; no fitting or cutoff-dependent constant enters.

Independent checks on a regional calculation

Relative entropy supplies checks that do not rely on knowing the answer in advance. If $B_1\subset B_2$ and both restrictions use the same state pair, data processing gives

$$D(\rho_{B_1}\Vert\sigma_{B_1}) \leq D(\rho_{B_2}\Vert\sigma_{B_2}).$$

This is a check on the complete combination $\Delta K-\Delta S$, not on either term separately: changing the region also changes its modular Hamiltonian. A numerical calculation that finds positive relative entropy on each region but violates monotonicity has probably mismatched its cutoffs, reference states, or region maps.

Three further controls isolate common mistakes. First, add an arbitrary constant $a\mathbf1$ to $K_\sigma$; both expectation values shift by $a$, so $\Delta\langle K_\sigma\rangle$ must remain unchanged. Second, send the excitation amplitude continuously to zero; the ratio between $\Delta S$ and $\Delta K$ should approach one at leading order, while their difference should vanish at least quadratically. Third, increase the regulator resolution while holding the physical region and state preparation fixed; $D$ should converge even though the two raw entropies grow. None of these tests proves that the chosen continuum limit lies in the theorem domain, but failure of any one decisively falsifies the implementation.

The support condition also has operational content. If a regulated reference assigns zero probability to a subspace populated by $\rho$, then no finite modular-energy difference can compensate for that perfectly distinguishable event: $D=+\infty$. Treating such a case as a finite saturated bound is a truncation error, not an exotic violation.

Directionality is another useful control. Relative entropy is not symmetric, so exchanging $\rho$ and $\sigma$ changes both the support condition and the modular Hamiltonian. Positivity of $D(\sigma_B\Vert\rho_B)$ therefore gives a second, different inequality involving $K_\rho=-\log\rho_B$; it does not sharpen the displayed bound by simple averaging. Likewise, $\Delta S_B$ need not be positive. A state can be more distinguishable from the reference while having a smaller regional entropy, in which case the upper bound remains valid but is not a statement that excitation always creates entropy. Code that silently replaces $\Delta S_B$ by $|\Delta S_B|$, or that inserts $K_\rho$ on the right while retaining the original ordering, is testing a different and generally false claim.

What the inequality does and does not say

Same algebra and same reference. Changing the region, reference state, or edge-mode convention changes the problem. A thermal reference has a different modular Hamiltonian from the vacuum reference.

Finite comparison, not finite absolute entropy. The algebraic relative entropy can exist even when neither entropy in $\Delta S_B$ exists separately. A regulator calculation must use the same geometry and subtraction on both states.

Regional QFT, not universal gravity. This page assumes a fixed background and a known reference modular operator. Black-hole area terms, backreaction, and the generalized second law require additional gravitational hypotheses.

No ordinary-energy replacement by inspection. The modular weight, operator content, and energy-density sign must be controlled. Generic regions have nonlocal modular Hamiltonians, so there may be no stress-tensor-only formula at all.

Decisive counterchecks. Move a fixed-energy packet toward $\partial B$, introduce a locally negative stress profile, or change the reference while retaining the old weight. Each operation defeats a naive $ER$ inference while leaving relative-entropy positivity intact.

Exercises

1. Derive the ball’s ordinary-energy corollary under a pointwise sign assumption.

Solution

Inside the ball, $w(r)=(R^2-r^2)/(2R)$ decreases from $R/2$ to $0$. If $\Delta\langle T_{00}\rangle\geq0$, then $\int w\,\Delta T_{00}\leq(R/2)\int\Delta T_{00}$. Multiplication by $2\pi$ gives $\Delta K_B\leq\pi R\Delta E_B$, and positivity gives $\Delta S_B\leq\pi R\Delta E_B$. The proof fails at exactly the step that multiplies the pointwise inequality by a sign-indefinite energy density.

2. Verify the thermal modular energy in the benchmark.

Solution

For constant energy density,

$$\int_{-R}^{R}dx\,\frac{R^2-x^2}{2R}=\frac{2R^2}{3}.$$

Therefore $\Delta K_B=2\pi(\pi c/6\beta^2)(2R^2/3)=2\pi^2cR^2/(9\beta^2)=cz^2/18$.

3. Show explicitly why the first law does not imply finite-state saturation.

Solution

For $\rho(s)=\sigma+s\,\delta\rho+O(s^2)$, positivity and $D(\sigma\Vert\sigma)=0$ force the first derivative of $D$ at $s=0$ to vanish. Since $D=\Delta K-\Delta S$, the linear terms agree. The quadratic term is the relative-entropy curvature and is generally positive, so equality need not persist at finite $s$.

4. Compare two narrow positive packets of equal energy at $x_0=0$ and $x_0=0.9R$.

Solution

The central packet has $\Delta K\simeq\pi ER$. The displaced packet has $\Delta K\simeq\pi ER(1-0.9^2)=0.19\pi ER$. Their ordinary energies agree, but the second modular energy is smaller by a factor of $0.19$. Thus total energy alone cannot reconstruct the exact bound.

References

• Blanco, David D., Horacio Casini, Ling-Yan Hung, and Robert C. Myers. “Relative Entropy and Holography.” Journal of High Energy Physics 2013, no. 8 (2013): 060. DOI. Open paper.
• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics: Theory and Experiment 2004, no. 6 (2004): P06002. DOI. Open paper.
• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25, no. 20 (2008): 205021. DOI. Open paper.
• Casini, Horacio, Marina Huerta, and Robert C. Myers. “Towards a Derivation of Holographic Entanglement Entropy.” Journal of High Energy Physics 2011, no. 5 (2011): 036. DOI. Open paper.