Quantum Focusing Condition (QFC)

Updated 4 October 2025

QFC is a conjectured inequality that extends the classical focussing theorem by incorporating quantum effects via a generalized entropy formulation.
It employs functional derivatives of generalized entropy to define quantum expansion, leading to quantum energy conditions like the QNEC and refined entropy bounds.
QFC underpins holographic and gravitational thermodynamic proofs, offering insights into black hole information, quantum singularity theorems, and entropy-area relations.

The Quantum Focusing Condition (QFC) is a conjecture in semiclassical gravity and quantum field theory that generalizes the classical focusing theorem to account for quantum effects in spacetime. The QFC is formulated as a universal, local inequality for the quantum expansion associated with a surface and its deformations along null congruences, based on the generalized entropy that combines the area term and the von Neumann entropy of quantum fields. Its implications are profound, leading to quantum generalizations of energy conditions, entropy bounds, and underpinning core consistency relations in holography and gravitational thermodynamics.

1. Generalized Entropy and Quantum Expansion

For a codimension-2 surface $\sigma$ , the generalized entropy is defined as: $S_{\text{gen}} = \frac{\text{Area}(\sigma)}{4G\hbar} + S_{\text{out}}$ where $\text{Area}(\sigma)$ is taken in Planck units, and $S_{\text{out}}$ is the von Neumann entropy of quantum fields outside $\sigma$ . Both terms are individually cutoff-dependent— $S_{\text{out}}$ possesses ultraviolet divergences matched by renormalization in the gravitational sector—but $S_{\text{gen}}$ itself is finite and physically meaningful.

Given a null hypersurface $N$ orthogonal to $\sigma$ , a local quantum expansion $\Theta$ is defined using functional derivatives of $S_{\text{gen}}$ : $\Theta[V(y); y_1] = \lim_{\mathcal{A}\to0}\frac{4G\hbar}{\mathcal{A}}\frac{dS_{\text{gen}}}{d\epsilon}|_{y_1}$ or equivalently as

$\Theta[V(y); y_1] = \frac{4G\hbar}{\sqrt{g|_{V}(y_1)}} \frac{\delta S_{\text{gen}}}{\delta V(y_1)}$

where the deformation $V(y)$ is localized in a neighborhood of $y_1$ and $\mathcal{A}$ is a transverse area element. In the classical limit, $\Theta$ reduces to the expansion $\theta$ of null geodesics.

2. Statement and Structure of the Quantum Focusing Conjecture

The QFC asserts that, under infinitesimal deformations of $\sigma$ along $N$ , the quantum expansion cannot increase in the null direction: $\left(\frac{\delta}{\delta V(y_2)}\right)\Theta[V(y); y_1] \leq 0$ This nonpositivity mirrors the classical focussing theorem (where $d\theta/d\lambda \leq 0$ ); however, the QFC is robust even in the presence of quantum matter that violates the classical null energy condition (NEC). The condition is made precise by using the second functional derivative of $S_{\text{gen}}$ .

3. Quantum Bousso Bound and Quantum Null Energy Condition (QNEC)

By integrating the QFC along a finite deformation, one obtains the quantum version of the Bousso bound: $S_{\text{gen}}[V(y)] \leq S_{\text{gen}}[0]$ or, explicitly: $S_{\text{out}}[\sigma'] + \frac{A(\sigma')}{4G\hbar} \leq S_{\text{out}}[\sigma] + \frac{A(\sigma)}{4G\hbar}$ which, upon rearrangement, gives

$\Delta S_{\text{out}} \leq \frac{\Delta A}{4G\hbar}$

This statement is cutoff independent and combines both matter and geometric entropy.

In the special case where the classical expansion and shear vanish, QFC yields a field-theoretic Quantum Null Energy Condition: $T_{kk} \geq \frac{\hbar}{2\pi \mathcal{A}} S_{\text{out}}''$ where $T_{kk}$ is the null-null component of the stress tensor and $S_{\text{out}}''$ is the second derivative of the von Neumann entropy. The QNEC provides a lower bound on $T_{kk}$ in terms of quantum information properties, and remains valid even when NEC violations occur.

4. Mathematical Formulation and Proof Structure

Key formulas:

Generalized entropy: $S_{\text{gen}} = \frac{A}{4G\hbar} + S_{\text{out}}$
Quantum expansion: $\Theta[V(y); y_1] = \frac{4G\hbar}{\sqrt{g|_{V}(y_1)}} \frac{\delta S_{\text{gen}}}{\delta V(y_1)}$
QFC: $\left(\frac{\delta}{\delta V(y_2)}\right)\Theta[V(y); y_1] \leq 0$
QNEC: $T_{kk} \geq \frac{\hbar}{2\pi\mathcal{A}}S_{\text{out}}''$

The proof sketch for the QFC uses strong subadditivity of the von Neumann entropy for off-diagonal variations, while the diagonal component is addressed via the Raychaudhuri equation supplemented by quantum corrections. In the latter, the quantum expansion is written as $\Theta = \theta + (4G\hbar/\mathcal{A})S_{\text{out}}'$ , and differentiation yields contributions from both geometry and entropy, leading to: $\Theta' = -8\pi G \langle T_{kk} \rangle + \left(\frac{4G\hbar}{\mathcal{A}}\right)S_{\text{out}}''$ Ensuring $\Theta' \leq 0$ derives the QNEC. The proof invokes perturbative null quantization and the replica trick.

5. Role in Holography and Quantum Gravity Theorems

Beyond energy conditions and entropy bounds, QFC forms the basis for numerous quantum gravity theorems, including:

Precise entropy bounds in holographic contexts (quantum Bousso bound)
Quantum singularity theorems, replacing classical conditions with quantum counterparts
Validity of entropy-area prescriptions in AdS/CFT via extremization arguments

Quantum extremal surface constructions, used for bulk reconstruction and entanglement wedge consistency, rely on the QFC to ensure correct causal and nesting properties. The conjecture also underlies recent progress on the black hole information paradox, where quantum effects must be balanced against classical area decrease—often utilizing the "island rule" for entropy calculations.

6. Extensions, Limitations, and Resolution of Apparent Violations

Apparent violations of the QFC arise in higher curvature gravity (e.g., Gauss-Bonnet terms in $d \geq 5$ ) (Fu et al., 2017). Such violations are associated with ill-defined pointwise evaluation of geometric terms. Correctly interpreted as expectation values in an effective field theory and properly smeared over regions larger than the cutoff scale (often Planck scale), these violations are removed (Leichenauer, 2017). Smearing is essential for consistency, as demonstrated in the analysis of entanglement wedge nesting in holography.

Recent developments have led to a restricted version of the QFC that only requires non-positive quantum expansion derivative when the expansion vanishes, which is sufficient for essential theorems in semiclassical gravity and has been proven in brane-world scenarios (Shahbazi-Moghaddam, 2022).

The QFC may face limitations when applied to certain coarse-grained entropy-like quantities (e.g., causal holographic information), which can violate the linearized QFC near caustics, though the generalized second law remains intact (Fu et al., 2018).

7. Contemporary Directions and Open Questions

Current research focuses on:

Extensions beyond the von Neumann case to discrete and conditional max/min entropy formulations, removing redundancies and resolving ambiguities at non-smooth points (Bousso et al., 23 Oct 2024).
Improved quantum energy conditions (INEC) that strengthen the QNEC by including first and second entropy derivatives (Ben-Dayan, 2023).
Demonstration of QFC validity across dynamical scenarios, including evaporating black holes and Page curve analyses (Matsuo, 2023, Ishibashi et al., 28 Mar 2024).

These efforts continue to clarify the link between quantum information theory, gravitational entropy, and spacetime dynamics, solidifying the QFC’s role in the axiomatic structure of quantum gravity.

In summary, the Quantum Focusing Condition strengthens the foundational interplay between quantum information and spacetime geometry, extends classical energy and entropy bounds to quantum regimes, and ensures the consistency and robustness of semiclassical gravitational theory and holographic dualities.