Linearized focusing and quantum entropy inequalities — Examples

Use natural units $c=\mathrm{\hslash }=k_B=1$. For each source formulation below, supply its own Hausdorff, second-countable smooth $d$-manifold $M$ without boundary, $d\ge 2$, with a smooth symmetric nondegenerate metric $g$ of signature $(-,+,\dots ,+)$ and a smooth timelike field $T$ fixing the future cone. A nonzero tangent vector $v$ is timelike, null or spacelike according as $g(v,v)<0$, $=0$ or $>0$; it is causal in the first two cases and future directed when $g(v,T)<0$. Use the metric-compatible torsion-free connection $\mathrm{\nabla }$, whose coordinate coefficients are ${\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{g}^{ae}({\mathrm{\partial }}_b{g}_{ec}+{\mathrm{\partial }}_c{g}_{eb}-{\mathrm{\partial }}_e{g}_{bc})$, with repeated indices summed from $0$ to $d-1$ and $(g^{ab})=(g_{ab}{)}^{-1}$. A causal curve is a continuous map from a real interval into $M$, locally piecewise $C^1$, with nonzero consistently directed causal tangent on each smooth piece. It is inextendible if its image admits no continuation as such a curve at either end. A Cauchy surface here is a smooth embedded spacelike hypersurface met in exactly one point by every inextendible causal curve; spacelike means that the induced metric is positive definite. A smooth embedded codimension-two cut splits such a surface when the complement is the disjoint union of two open sides whose common boundary is the cut. Supply a quantum-field state and its stress expectation $⟨T_{ab}⟩$ on the modeled domain, with the source's renormalization and differentiability conditions. These data describe the cited setups, independently of the four-dimensional linearized horizon model.

For the cited quantum null energy condition (QNEC), let $\mathrm{\Sigma }\subset M$ be such a cut and let $p\in \mathrm{\Sigma }$. Let $\mathcal{N}$ be a smooth null hypersurface orthogonal to $\mathrm{\Sigma }$ near $p$, with smooth future-directed null generators labelled by transverse coordinates $y$. On each generator use an affine parameter $\lambda$ increasing toward the future with $[\lambda ]=L$, and normalize its tangent by $k^a=(d\mathrm{/}d\lambda {)}^a$. Deforming $\mathrm{\Sigma }$ to $\mathrm{\Sigma }[\lambda (y)]$ moves each generator by its indicated affine amount. Restrict to a smooth embedded null hypersurface segment before generator intersections: the radical $L_p:=\{v\in T_p\mathcal{N}:g(v,w)=0\text{ for all }w\in T_p\mathcal{N}\}$ is one-dimensional, is spanned by $k_p$, and ${\mathrm{\nabla }}_{k}k=0$. For $d>2$, in transported cut coordinates put $B_{AB}:=g({\mathrm{\nabla }}_{{\mathrm{\partial }}_{y^A}}k,{\mathrm{\partial }}_{y^B})$, $\theta :=q^{AB}{B}_{AB}$ and ${\sigma }_{AB}:=B_{AB}-\theta q_{AB}\mathrm{/}(d-2)$, where $q$ is the induced positive cut metric. These define the classical expansion and shear; there are no transverse expansion or shear components when $d=2$. The source's $S_{\mathrm{o}\mathrm{u}\mathrm{t}}[\lambda (y)]$ is the von Neumann entropy of the quantum-field state restricted to one side of the deformed cut, equivalently to the corresponding side of a Cauchy surface; the entropy itself may be divergent, while the source-defined derivatives in the inequality are finite. With a regulator admitting a positive trace-one reduced density operator $\rho$, the von Neumann entropy is $-\mathrm{Tr}(\rho \log \rho )$, with $0\log 0=0$; continuum entropy and its derivatives use the cited limiting prescription, not an assumed finite unregulated density matrix. In §2, equation (2.2), the authors call the pointwise formulation the “general conjecture”; its local geometric condition is that the classical expansion and shear vanish at $p$. For $d>2$ and a transverse patch of area $A$ about $p$, $S_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{\prime }\mathrm{\prime }}(p)$ is the source-defined second affine variation for that patch, with the other generators held fixed. The corresponding pointwise formulation, in the small-area notation of equation (2.3), is $⟨T_{kk}(p)⟩\ge \frac{1}{2\pi }{\lim }_{A\to 0}\frac{S_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{\prime }\mathrm{\prime }}(p)}{A},⟨T_{kk}(p)⟩:=⟨T_{ab}(p)k^a{k}^b⟩\mathrm{.}$ Here $A$ is the geometric area of the transverse patch, computed from the Riemannian metric induced on $\mathrm{\Sigma }$, so $[A]=L^{d-2}$; both sides then have dimension $L^{-d}$. Separately, the paper's abstract and outline state applicability to free or superrenormalizable, massive or massless bosonic fields on stationary null hypersurfaces, where “stationary” means that the classical expansion vanishes everywhere. The authors explicitly treat the QNEC proof as a fixed-background calculation without dynamical gravity. Its §3.1–§4.5 argument treats $D>2$ on such a hypersurface. Section 5 derives the free-scalar $D=2$ inequality $⟨T_{2D}⟩\ge (2\pi {)}^{-1}{S}_{2D}^{\mathrm{\prime }\mathrm{\prime }}$ by dimensional reduction and strong subadditivity, with no shrinking transverse-area limit; there $T_{2D}$ is the stress tensor of the reduced chiral scalar, whose action on the reduced theory agrees with the transverse integral of $T_{kk}$, and $S_{2D}^{\mathrm{\prime }\mathrm{\prime }}$ is the reduced entropy's second affine variation. Section 5 also treats higher-spin bosonic fields and discusses extending the result to superrenormalizable interactions conditionally on preservation of the null-surface observable algebra and $T_{kk}$; the paper presents that preservation as expected, not as established for arbitrary such interactions. These source-reported scopes are distinct from the general pointwise conjectural formulation of §2; they do not establish that formulation for every field theory and background.

For the quantum focusing conjecture (QFC) in the cited semiclassical Einstein-gravity formulation, use the geometry above and let $\sigma$ be a smooth spacelike codimension-two cut splitting a Cauchy surface, and let $\mathcal{N}$ be the smooth null hypersurface generated by its orthogonal future-directed null geodesics. Label the generators by local transverse coordinates $y=(y^1,\dots ,y^{d-2})$ in $d$ spacetime dimensions. On each generator choose an affine parameter $\lambda$ increasing to the future, with $\lambda =0$ at $\sigma$, $[\lambda ]=L$, and tangent $k=(d\mathrm{/}d\lambda )$. A smooth nonnegative function $V(y)$ defines a cut $\sigma [V]$ by its intersection with generator $y$ at $\lambda =V(y)$; restrict to profiles for which this is a well-defined smooth cut that still splits a Cauchy surface. Let $q_{AB}[V]$ be the positive induced metric on $\sigma [V]$ in the coordinates $y$, set $h[V](y):=\det (q_{AB}[V](y))$, and define its area by $A[V]:={\int }_{\sigma [V]}\sqrt{h[V](y)}{d}^{d-2}y\mathrm{.}$ For $d=2$, a cut is discrete, the empty metric determinant is $1$, and this integral is counting measure; functional derivatives are derivatives with respect to the individual cut-point displacements. The JT formulation below uses a different gravitational entropy term and normalization. Let $S_{\mathrm{o}\mathrm{u}\mathrm{t}}[V]$ be the von Neumann entropy of the quantum-field state restricted to one side of $\sigma [V]$. It may be ultraviolet divergent by itself. The cited source defines the generalized entropy by the following combination and assumes its renormalized value is finite and cutoff-independent: $S_{\mathrm{g}\mathrm{e}\mathrm{n}}[V]:=\frac{A[V]}{4G}+S_{\mathrm{o}\mathrm{u}\mathrm{t}}[V]+S_{\mathrm{c}\mathrm{t}}[V],$ where $S_{\mathrm{c}\mathrm{t}}[V]$ is the source-prescribed local geometric counterterm contribution; this is the source's $S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}[V]=A[V]\mathrm{/}(4G)+S_{\mathrm{c}\mathrm{t}}[V]$ plus $S_{\mathrm{o}\mathrm{u}\mathrm{t}}[V]$, with $\mathrm{\hslash }=1$. In $d$ dimensions, $G>0$ has dimension $L^{d-2}$, so $S_{\mathrm{g}\mathrm{e}\mathrm{n}}$ is dimensionless. The source defines the quantum expansion as the entropy response to a localized affine displacement, divided by the geometric area of the transverse patch; in its functional-derivative convention this is $\mathrm{\Theta }[V;y_1]:=\frac{4G}{\sqrt{h[V](y_1)}}\frac{\delta S_{\mathrm{g}\mathrm{e}\mathrm{n}}[V]}{\delta V(y_1)}\mathrm{.}$ For every admissible one-parameter family $V_{\tau }$ with ${\mathrm{\partial }}_{\tau }{V}_{\tau }(y)\ge 0$, the QFC conjectures $\frac{\delta \mathrm{\Theta }[V;y_1]}{\delta V(y_2)}\le 0$ for all generator labels $y_1,y_2$, with the coincident-point diagonal understood in the source's distributional sense. This remains a conjecture in general.

The cited 2017 Einstein-Hilbert-Gauss-Bonnet analysis constructs weak-curvature QFC violations for $d\ge 5$ at points where the null expansion and shear vanish. The construction uses arbitrarily fine transverse localization. The authors' v2 note reports that restricting entropy variations to surfaces smooth on the scale associated with $G\gamma$ removes their violation, and leaves the effective-field-theory-scale formulation as an open issue.

The cited 2026 v3 paper proves the restricted QFC (rQFC) for a class of two-dimensional Jackiw-Teitelboim (JT) gravity models coupled to quantum field theory, in regions where the total dilaton is positive and the matter state satisfies the source's two-dimensional QNEC. The total dilaton is the smooth scalar $\mathrm{\Phi }={\varphi }_0+\varphi :M\to \mathbb{R}$ of that two-dimensional model, with constant ${\varphi }_0$; it replaces transverse area in the entropy term. For a moving cut point $P(\lambda )$ on a future affine null geodesic, $\lambda \in I\subset \mathbb{R}$, keep the other cut points fixed and use $S_{\mathrm{g}\mathrm{e}\mathrm{n}}={\sum }_i\mathrm{\Phi }(P_i)\mathrm{/}(4G)+S_{\mathrm{r}\mathrm{e}\mathrm{n}}$, where $S_{\mathrm{r}\mathrm{e}\mathrm{n}}$ is the source-renormalized matter entropy. Here $G>0$ and $\mathrm{\Phi }$ are dimensionless, and the source's quantum expansion is $\mathrm{\Theta }(\lambda )=4G\mathrm{\Phi }(P(\lambda ){)}^{-1}d{S}_{\mathrm{g}\mathrm{e}\mathrm{n}}\mathrm{/}d\lambda$, with $\mathrm{\Phi }(P(\lambda ))>0$ and the derivatives assumed to exist. Its rQFC asserts $d\mathrm{\Theta }\mathrm{/}d\lambda \le 0$ where $\mathrm{\Theta }=0$; this is weaker than requiring that inequality at every cut. The authors also construct full-QFC counterexamples in a JT regime where matter quantum effects compete with the total dilaton. They state that this regime has no higher-dimensional uplift; although the JT action has an approximate dimensional-reduction origin, they treat this counterexample regime as an exact two-dimensional toy model and caution that it is not strong evidence against the original QFC. These theory- and regime-specific findings establish no universal QFC theorem for arbitrary higher-curvature effective gravity. The linearized horizon model discussed here defines no outside entropy, generalized-entropy functional, or entropy shape derivative; it neither assumes nor derives QNEC or QFC.

Linearized future horizon focusing model

Bousso, Fisher, Koeller, Leichenauer and Wall, Proof of the Quantum Null Energy Condition

Bousso, Fisher, Leichenauer and Wall, A Quantum Focussing Conjecture

Fu, Koeller and Marolf, Violating the Quantum Focusing Conjecture and Quantum Covariant Entropy Bound in d ≥ 5 dimensions

Franken et al., Tests of restricted quantum focusing and a new CFT bound

The QNEC and QFC claims above are reported source formulations, not local proof premises. The higher-curvature result is limited to the models and dimensions in its cited analysis; its discussion also distinguishes the pointwise conjecture from possible effective-field-theory resolution conditions.