Quantum Focusing and Generalized Expansions

Classical focusing tracks the area expansion of a null congruence. Quantum focusing replaces area by renormalized generalized entropy and asks whether its expansion can increase under further forward null deformation. The unrestricted quantum focusing conjecture (QFC) is not a universal theorem: pointwise higher-curvature counterexamples in dimensions $d\ge5$ require an EFT-resolution qualification, and controlled two-dimensional JT models supply a different failure of the unrestricted inequality. A restricted version imposes monotonicity where quantum expansion vanishes; its general validity remains open.

Evidence cutoff. This assessment was checked through 2026-09-26. The JT and CFT analysis of Franken and collaborators is published in Journal of High Energy Physics 05 (2026), 111; the cited arXiv copy is version 3 of 2026-06-29.

Required background. Congruences and Optical Scalars supplies classical expansion, shear, and Raychaudhuri evolution, and Quantum Information supplies entropy and strong subadditivity.

Helpful background. Quantum Fields in Curved Spacetime supplies renormalization, Einstein Equations supplies backreaction, and Black-Hole Geometry and Mechanics supplies null hypersurfaces and horizons.

Generalized entropy and quantum expansion

Let $\sigma$ be a smooth codimension-two surface that splits a Cauchy surface. Choose one side and a null hypersurface $N$ orthogonal to $\sigma$. A cut of $N$ is specified by an affine displacement $V(y)$ along each generator. Its generalized entropy is

$$S_{\mathrm{gen}}[V] =S_{\mathrm{grav}}[V]+S_{\mathrm{out}}[V],$$

where

$$S_{\mathrm{grav}} =\frac{A[V]}{4G} +\text{local higher-curvature entropy terms}.$$

The same regulator and renormalized couplings must make the sum finite. The quantum expansion at generator $y_1$ is

$$\Theta[V;y_1] =\frac{4G}{\sqrt{h(y_1)}} \frac{\delta S_{\mathrm{gen}}[V]}{\delta V(y_1)}.$$

Unlike the classical expansion $\theta$, $\Theta$ is state dependent and nonlocal in the entire cut. Bousso and collaborators define $S_{\mathrm{gen}}$, the localized deformation, and $\Theta$ in Bousso et al. 2016, §§ 3.1–3.2, Eqs. (3.1)–(3.9), PDF.

The unrestricted QFC states

$$\frac{\delta\Theta[V;y_1]}{\delta V(y_2)}\le0$$

for forward deformation at any pair of generators $y_1,y_2$. The restricted QFC (rQFC) requires this inequality at points where $\Theta[V;y_1]=0$. The latter is weaker but is argued to preserve the sign-crossing implication used in many applications. Its precise proposal and holographic braneworld domain are given in Shahbazi-Moghaddam 2024, §§ II–IV, PDF.

Results and open questions through 2026-09-26

The evidence cutoff for this matrix is 2026-09-26. “Quantum focusing” combines distinct theorems, conjectures, controlled limits, and counterexamples. Dimension, gravitational action, and transverse resolution must accompany a diagonal finite-gravity claim.

Evidence or result	Assumptions and formulation	Conclusion	Limitation or unresolved boundary
Original QFC analysis, Bousso et al. 2016, §§ 3–6, PDF	Renormalized generalized entropy on smooth Cauchy-splitting cuts; semiclassical expansion	Proposes full functional monotonicity; proves the off-diagonal part from strong subadditivity; derives QNEC as a diagonal weak-gravity limit.	The diagonal finite-$G$ QFC is conjectural, not proved by the QNEC limit.
QNEC proofs, Bousso et al. 2016, §§ 2–5, PDF and Balakrishnan et al. 2019, §§ 1–7, PDF	Stationary free/superrenormalizable setting, then broad interacting Minkowski QFT with modular assumptions	Establishes the nongravitational diagonal limit in substantial QFT classes.	Does not establish finite-gravity focusing for arbitrary curved cuts.
Higher-curvature counterexample and resolution, Fu, Koeller, and Marolf 2017, § 2, Eqs. (2.17)–(2.19), PDF, and Leichenauer 2017, §§ III–IV, PDF	Weak curvature, nonzero Gauss–Bonnet coupling, $d\ge5$; distinguish a pointwise cut from averaging over an EFT-sized transverse region	The pointwise QFC can fail for either coupling sign; the proposed smearing removes this leading violation.	Neither an unrestricted pointwise theorem nor a proof of all smeared QFC formulations follows.
Restricted QFC, Shahbazi-Moghaddam 2024, §§ II–IV, PDF	Monotonicity imposed at $\Theta=0$; proof in a class of holographic braneworld semiclassical theories	Supplies a weaker condition argued sufficient for known sign-preservation applications.	General semiclassical validity outside the proved holographic class remains conjectural.
JT-gravity tests, Franken et al. 2026, §§ 2–3, PDF	Large-species semiclassical JT gravity coupled to QFT, treated as a two-dimensional model	Proves rQFC in the analyzed model and constructs original-QFC violations when matter effects compete with the total dilaton scale.	The violating hierarchy has no known higher-dimensional uplift; publication does not remove that model restriction.
New CFT consequence, Franken et al. 2026, § 4 and Appendix B, PDF	Near-vacuum states in $d\gt 2$ CFT; simultaneous weak-gravity and narrow-profile limit	rQFC implies a bound stronger than QNEC, forbidding too-fast saturation in the analyzed class.	A universal strengthened-QNEC extension is speculative and lacks an independent proof.

The matrix supports the off-diagonal theorem and QNEC limits strongly. It supports rQFC in specific holographic and two-dimensional settings. The pointwise higher-curvature and JT counterexamples must not be combined into an assertion that every resolution-aware formulation fails, nor omitted to suggest a generic finite-gravity theorem.

Established limits and independent checks

For $y_1\ne y_2$, local geometric counterterms have no off-diagonal functional derivative. The relevant entropy combination is

$$S(B)-S(BC)-S(AB)+S(ABC)\le0,$$

which is strong subadditivity. This proves off-diagonal QFC, with the ultraviolet boundary terms canceling in the same combination, as shown in Bousso et al. 2016, § 5.1, Eqs. (5.2)–(5.5), PDF.

For a diagonal deformation in $d\gt 2$ dimensions, suppress higher-curvature terms only in a regime where that is justified. In this Einstein-gravity approximation,

$$\Theta =\theta+\frac{4G}{\sqrt h}S_{\mathrm{out}}',$$

and Raychaudhuri plus the semiclassical equation gives schematically

$$\Theta' =-\frac{\theta^2}{d-2} -\sigma_{ab}\sigma^{ab} -8\pi G\langle T_{kk}\rangle +\frac{4G}{\sqrt h} \left(S_{\mathrm{out}}''-\theta S_{\mathrm{out}}'\right).$$

At a locally stationary point with $\theta=\sigma_{ab}=0$, QFC implies

$$\langle T_{kk}\rangle \ge\frac{1}{2\pi\sqrt h}S_{\mathrm{out}}'',$$

which is QNEC. This is an independent normalization and limiting check, but logical implication runs only from full QFC to QNEC; proofs of QNEC do not reverse into a proof of QFC.

In the classical limit with matter entropy suppressed, $\Theta\to\theta$. The conjecture then reduces to classical focusing when the classical null-energy and gravitational equations hold. If higher-curvature entropy terms are comparable, they must be retained simultaneously in $S_{\mathrm{grav}}$ and the field equations.

Competing interpretations

Unrestricted position. The full QFC is the natural generalized-entropy replacement for Raychaudhuri focusing. Its off-diagonal theorem, QNEC consequence, quantum Bousso-bound implication, and successful uses suggest a common principle rather than unrelated inequalities.

Restricted position. Known applications usually need only that a nonpositive quantum expansion cannot cross to positive values. rQFC captures that sign barrier, has proofs in specific holographic constructions, and survives the 2026 JT tests that violate the stronger QFC.

Conservative position. Only the off-diagonal component and particular diagonal limits are established broadly. The higher-curvature pointwise counterexamples require explicit EFT smearing, while the JT examples test a different model and scale hierarchy. The new CFT bound is a prediction of rQFC, not an independent proof of it.

These positions differ in field content, background dimension, regulator, state, solution branch, and semiclassical scaling. Calling every entropy inequality “QFC evidence” would erase those axes.

Limitations and adverse regime

Generalized entropy is meaningful only after geometric counterterms cancel the entropy divergences. A cutoff-dependent $S_{\mathrm{out}}''$ cannot be inserted into $\Theta$ alone. Caustics bound the admissible deformation family, and large metric fluctuations undermine the classical background needed to define its null generators.

There are two different adverse tests. In weakly curved $d\ge5$ Einstein–Gauss–Bonnet gravity, the classical area and higher-curvature entropy terms can yield a positive diagonal variation at a point with vanishing expansion and shear. The sign can be positive for either sign of the coupling. Transverse averaging restores negative expansion/shear contributions over a cutoff-sized region and removes this leading violation in Leichenauer’s analysis. The relevant distinction is pointwise versus EFT-resolved focusing, not merely small versus large curvature; see Fu, Koeller, and Marolf 2017, §§ 2 and 4, PDF and Leichenauer 2017, § IV, Eqs. (12)–(17), PDF.

In the JT examples, the gravitational variable replacing transverse area is the total dilaton $\Phi=\phi_0+\phi$. Work in a region with $\Phi\gt 0$, let $G_2$ be the two-dimensional Newton coupling, and vary one endpoint along an affine null parameter while holding the other endpoints fixed. The endpoint contributes $\Phi/(4G_2)$ to gravitational entropy, so the relevant expansions are

$$\theta=\frac{\Phi'}{\Phi}, \qquad \Theta=\frac{\Phi'+4G_2S_{\mathrm{out}}'}{\Phi}.$$

Here $\theta$ is the dilaton analogue of area expansion; it is not obtained by setting $d=2$ in the Raychaudhuri formula above. The JT null equation $\Phi''=-8\pi G_2\langle T_{kk}\rangle$ gives the useful identity

$$\Theta'+\theta\Theta =\frac{4G_2}{\Phi} \left(S_{\mathrm{out}}''-2\pi\langle T_{kk}\rangle\right).$$

Thus the two-dimensional QNEC implies

$$\Theta'\le-\theta\Theta.$$

Hence $\Theta=0$ implies $\Theta'\le0$, proving the stated rQFC in that model. This argument requires positive $\Phi$ and matter for which the stated two-dimensional QNEC holds, including the CFT and holographic QFT settings analyzed in Franken et al. 2026, § 3, Eqs. (3.3), (3.7), and (3.9)–(3.15), PDF.

Regions with $\theta\Theta\lt0$ permit violation of the stronger demand $\Theta'\le0$. The inequality alone permits a violation; the paper’s explicit states establish that it occurs. The examples require the species/backreaction scale to compete with the total dilaton. The usual higher-dimensional reduction hierarchy instead preserves full QFC in the analyzed regime, so these are genuine two-dimensional examples rather than an uplifted counterexample; see Franken et al. 2026, § 3, Eqs. (3.16)–(3.20), PDF.

Open questions. The conclusions could change when a calculation reproduces or eliminates a violation in a new controlled model, a direct QFT argument proves or refutes the proposed stronger CFT bound, an EFT-smearing analysis changes the allowed focusing statement, or a central source is corrected or withdrawn. One valid counterexample can disprove a universal bound; agreement between independent calculations strengthens confidence in a proposed restricted bound.

Exercise

Starting from the diagonal expression for $\Theta'$, derive QNEC at a locally stationary point. Then explain why QNEC saturation does not prove full QFC.

Solution

Set $\theta=0$ and $\sigma_{ab}=0$ at the point. The mixed term $-\theta S_{\mathrm{out}}'$ and the quadratic geometric terms vanish, leaving

$$\Theta' =-8\pi G\langle T_{kk}\rangle +\frac{4G}{\sqrt h}S_{\mathrm{out}}''.$$

If QFC holds, $\Theta'\le0$. Dividing by $8\pi G\gt 0$ gives

$$\langle T_{kk}\rangle \ge\frac{1}{2\pi\sqrt h}S_{\mathrm{out}}'',$$

the QNEC normalization. Saturation only says that this particular locally stationary, diagonal, weak-gravity combination vanishes. Full QFC also covers finite gravitational entropy terms, nonstationary cuts, and variations at arbitrary pairs of generators. None of those data are fixed by QNEC saturation, so the converse implication fails.

References

• Balakrishnan, S., Faulkner, T., Khandker, Z. U., and Wang, H. 2019. “A General Proof of the Quantum Null Energy Condition.” Journal of High Energy Physics 2019, 020. DOI. Open PDF.
• Bousso, R., Fisher, Z., Koeller, J., Leichenauer, S., and Wall, A. C. 2016. “Proof of the Quantum Null Energy Condition.” Physical Review D 93, 024017. DOI. Open PDF.
• Bousso, R., Fisher, Z., Leichenauer, S., and Wall, A. C. 2016. “Quantum Focusing Conjecture.” Physical Review D 93, 064044. DOI. Open PDF.
• Franken, V., Kaya, S., Rondeau, F., Shahbazi-Moghaddam, A., and Tran, P. 2026. “Tests of Restricted Quantum Focusing and a New CFT Bound.” Journal of High Energy Physics 05 (2026), 111. DOI. Open PDF.
• Fu, Z., Koeller, J., and Marolf, D. 2017. “Violating the Quantum Focusing Conjecture and Quantum Covariant Entropy Bound in d ≥ 5 Dimensions.” Classical and Quantum Gravity 34, 175006. DOI. Open PDF.
• Leichenauer, S. 2017. “The Quantum Focusing Conjecture Has Not Been Violated.” arXiv:1705.05469. Open PDF.
• Shahbazi-Moghaddam, A. 2024. “Restricted Quantum Focusing.” Physical Review D 109, 066023. DOI. Open PDF.