Restricted Quantum Focusing Conjecture (rQFC)

• rQFC is a quantum gravity principle that refines QFC by enforcing a non-increasing quantum expansion only when the expansion vanishes.
• It has been rigorously proven in models like brane-world and JT gravity, ensuring consistency in semiclassical gravity and holography.
• The conjecture underpins improved energy bounds and entanglement properties, directly impacting QNEC and holographic entanglement wedge construction.

The Restricted Quantum Focusing Conjecture (rQFC) is a central principle in modern quantum gravity and holography. It refines the original Quantum Focusing Conjecture (QFC) by asserting a non-increasing condition on the quantum expansion only at points where the expansion vanishes, rather than everywhere along a null congruence. The rQFC streamlines the foundational structure of semiclassical gravity, providing a critical link between the behaviors of generalized entropy, energy conditions, and entropy bounds such as the Quantum Bousso bound and the Quantum Null Energy Condition (QNEC). The conjecture has been rigorously proven in certain semiclassical gravity contexts, notably brane-world scenarios and two-dimensional Jackiw–Teitelboim gravity coupled to quantum field theory. Recent research has established its sufficiency for the derivation of essential quantum gravity theorems, its necessity for consistent holographic entanglement wedge construction, and its direct relevance for energy constraints in conformal field theories.

1. Definition and Mathematical Formulation

The rQFC concerns the generalized entropy $S_{\text{gen}}$ of a codimension-2 surface, given by

$$S_{\text{gen}} = \frac{\text{Area}}{4G} + S_{\text{out}},$$

where "Area" is the geometric area (e.g., dilaton value in JT gravity for $d = 2$), and $S_{\text{out}}$ denotes the von Neumann entropy of quantum fields outside the surface. The quantum expansion $\Theta$ is defined via the functional derivative

$$\Theta_{(k)}[V;y] = \frac{4G}{\sqrt{h}} \frac{\delta S_{\text{gen}}}{\delta V(y)},$$

along null generators labeled by an affine parameter $\lambda$.

The central statement of the rQFC is: $$\Theta_{(k)}(V_{\lambda}; y) = 0 \quad \Longrightarrow \quad \frac{\partial}{\partial \lambda} \Theta_{(k)}(V_{\lambda}; y) \leq 0,$$ i.e., at any point where the quantum expansion vanishes, its variation under further null deformations is non-positive (Shahbazi-Moghaddam, 2022, Franken et al., 15 Oct 2025, Franken, 17 Oct 2025). This "restricted" form is weaker than the full QFC that demands non-positive variation everywhere, but captures all known necessary implications for semiclassical gravity.

2. Physical Interpretation and Scope

The rQFC generalizes the classical focusing theorem, which states that the geometric expansion $\theta$ of a null congruence satisfies

$\frac{d\theta}{d\lambda} \leq 0,$

when the null energy condition holds. Quantum effects can violate this classical condition due to negative energy densities (e.g., in Hawking radiation). The rQFC replaces the classical expansion with the quantum expansion $\Theta$, accommodating quantum violations of energy conditions while ensuring the generalized entropy's monotonicity, especially on quantum extremal surfaces where $\Theta = 0$.

Operationally, the rQFC is sufficient to derive:

• The Quantum Null Energy Condition (QNEC):

$$\langle T_{kk} \rangle \geq \frac{\hbar}{2\pi \mathcal{A}} S_{out}'' \quad \text{at } \theta = 0,$$

or its improved form (INEC) when $\Theta \rightarrow 0$ (Ben-Dayan, 2023).

• The Quantum Bousso bound:

$$S_{th}(\mathcal{L}_{qu}) \leq \left[ \text{Area}_{qu}(\gamma_1) - \text{Area}_{qu}(\gamma_2) \right]/(4G),$$

for lightsheets with non-positive quantum expansion (Shahbazi-Moghaddam, 2022, Franken, 17 Oct 2025).

The rQFC underpins singularity theorems, entropy bounds, holographic entanglement wedge nesting, and causality constraints in AdS/CFT.

3. Rigorous Proofs and Explicit Models

The rQFC has been rigorously justified in several frameworks:

• Brane-world semiclassical gravity: The proof relies on analyzing deformations of extremal surfaces in AdS bulk geometry, translating the problem into a cooperative elliptic system for the surface deformation parameters. Under mild technical assumptions (existence of a supersolution), the cooperative structure ensures non-positivity of the quantum expansion's derivative at points where it vanishes (Shahbazi-Moghaddam, 2022).
• Jackiw–Teitelboim gravity in $d=2$: With the generalized entropy defined as $S_{gen}(W) = (1/4G) \sum_i \Phi(P_i) + S_{ren}(W)$, and the quantum expansion $\Theta_\lambda = (4G/\Phi) (dS_{gen}/d\lambda)$, one derives the evolution equation $\Theta' \leq -\theta \Theta$, implying $\Theta' \leq 0$ at $\Theta = 0$ (Franken et al., 15 Oct 2025, Franken, 17 Oct 2025). This construction leverages the rigorous validity of the two-dimensional QNEC.
• Counter-examples to the full QFC: In both JT gravity and evaporating de Sitter horizons, explicit configurations are presented in which quantum corrections rival the geometric term (i.e., the species scale $\ell_S \sim c G$ is comparable to the dilaton), and the unrestricted QFC is violated, but the rQFC remains intact (Franken et al., 15 Oct 2025, Franken, 17 Oct 2025).

4. Implications for Energy Conditions, Holography, and Quantum Field Theory

The rQFC implies sharpened energy bounds in quantum field theory and holography:

• Improved Quantum Null Energy Condition (INEC): For smooth surfaces with vanishing quantum expansion,

$$T_{kk} \geq \frac{\hbar}{2\pi \mathcal{A}} \left( S_{out}'' - \frac{1}{2} \theta S_{out}' \right ),$$

furnishing a stronger bound than the standard QNEC, independent of Newton's constant G (Ben-Dayan, 2023).

• Bounds on operator spectrum in CFT: In $d > 2$, the rQFC constrains the short-distance behavior of the product of two averaged null energy (light-ray) operators, imposing an upper bound $\delta_0 \leq 2(d-1)$ on the scaling dimension of the leading even-spin Regge trajectory (Franken et al., 15 Oct 2025). This CFT bound is saturated in planar $\mathcal{N}=4$ SYM at strong coupling.
• Discrete nonexpansion and strong subadditivity: The discrete max-QFC reformulation achieves focusing via nonpositivity of the conditional max entropy for all outward null deformations, eliminating the need for a numerical expansion at caustics or corners (Bousso et al., 23 Oct 2024). This discrete version enables rigorous proofs of subadditivity, nesting, and complementarity for entanglement wedges in holography.
• Covariant entanglement prescription and subregion duality: In cosmological settings (de Sitter and FLRW), the rQFC informs the constrained extremization prescription for holographic entanglement entropy, ensuring that the causal structure and mutual information between regions encode the geometric connectivity of the bulk (Franken, 17 Oct 2025).

5. Technical Nuances: Smearing, Cutoffs, and Dimensional Dependence

The necessity of "smearing" the quantum expansion over scales at least as large as the cutoff length $\ell$ of the effective field theory is essential for avoiding apparent violations of the QFC induced by higher-curvature corrections such as Gauss–Bonnet terms (Fu et al., 2017, Leichenauer, 2017, Kanai et al., 2 May 2024). For instance, in quadratic gravity with dimension $d \geq 5$, the QFC is restored when the expansion is averaged over regions larger than $\ell_+ = 4\sqrt{6\gamma}$ or $\ell_- = \sqrt{48((3d-7)(d-4)/(d-2))|\gamma|}$, depending on the sign of the coupling (Kanai et al., 2 May 2024). This cutoff-dependent version effectively restricts the QFC to its rQFC form at the operational scale.

Contrasts between the validity of rQFC and violations of the unrestricted QFC are present especially when quantum corrections become large relative to geometric quantities (e.g., for large central charge or high matter species number), advocating the rQFC as the more robust semiclassical condition.

6. Applications, Causality, and Future Directions

The rQFC informs and governs:

• Quantum singularity theorems and generalized second law proofs,
• The rigorous construction of quantum extremal surfaces and entanglement wedges in AdS/CFT, including strong subadditivity and nesting properties (Bousso et al., 23 Oct 2024),
• Constraints on black hole evaporation: In both two- and higher-dimensional models, the rQFC is necessary for the existence of islands and reproduction of the expected Page curve; it manifests as geometric requirements (e.g., $f''(r) < 0$ near the horizon) ensuring the consistency of entropy evolution (Yu et al., 6 May 2024, Matsuo, 2023, Ishibashi et al., 28 Mar 2024).

A plausible implication is that further refinements, such as a strengthened QNEC incorporating the scaling behavior of entropy derivatives, may directly follow from the rQFC in CFTs (Franken et al., 15 Oct 2025).

Outstanding open directions include the systematic paper of rQFC in effective theories with multiple cutoffs, its extension to non-Lorentzian or non-EFT gravitational backgrounds, and an axiomatic classification of its discrete and one-shot quantum generalizations.

7. Summary Table: Comparison of QFC vs rQFC

Principle	Full QFC	Restricted QFC (rQFC)
Condition	$(\delta/ \delta V(y_2)) \Theta[V; y_1] \leq 0$ anywhere	$\Theta=0 \implies \partial_\lambda \Theta \leq 0$
Scope of Validity	All points along null congruence	Points (surfaces) of vanishing quantum expansion
Proven contexts	Limited; violations in strong quantum regimes, $d \geq 5$, etc.	Proven in brane-world, JT gravity, 2D evaporating BH
Implications	Implies QNEC, QBB, singularity theorems	Sufficient for QNEC, QBB, entanglement wedge nesting
Holographic significance	Foundational, but technically subtle in non-smooth regions	Robust under discrete and one-shot modifications

The Restricted Quantum Focusing Conjecture is an indispensable axiom for semiclassical gravity, quantum holography, and quantum information constraints in field theory. It anchors the consistency of entropy bounds, bulk reconstruction, and energy conditions beyond the classical regime and is poised to shape future developments in quantum gravity and CFT constraints.