The Quantum Null Energy Condition

The quantum null energy condition (QNEC) permits negative local null energy but bounds it by how entanglement entropy bends under a localized null deformation of an entangling cut. Its right-hand side is a diagonal second shape derivative, not an ordinary time derivative of the entropy of a fixed region. The original proof, broader interacting flat-space proofs, and curved-space formulations have different hypotheses and must not be merged into one unrestricted theorem.

Required background. Congruences and Optical Scalars supplies null generators, expansion, shear, and affine normalization, and Quantum Information supplies reduced states, von Neumann entropy, and modular Hamiltonians.

Helpful background. Quantum Fields in Curved Spacetime supplies renormalized stress tensors and states, Einstein Equations supplies later backreaction, and Black-Hole Geometry and Mechanics supplies stationary null surfaces.

The local entropy–stress inequality

Choose a point $p$, a future-directed null vector $k^a$ at $p$, and a smooth codimension-two cut $\Sigma$ orthogonal to the associated null congruence. Let $y$ label its generators and let $\lambda(y)$ be affine displacement along them. The cut divides a Cauchy surface, so $S_{\mathrm{out}}[\lambda(y)]$ is the entropy of the quantum fields on one chosen side.

The second functional derivative has diagonal and off-diagonal pieces:

$$\frac{\delta^2S_{\mathrm{out}}} {\delta\lambda(y)\delta\lambda(y')} =\frac{\delta^2S_{\mathrm{out}}}{\delta\lambda(y)^2} \delta^{(d-2)}(y-y') +\text{off-diagonal}.$$

Writing $h$ for the determinant of the induced metric on $\Sigma$, the QNEC at $p$ is

$$\boxed{ \langle T_{ab}k^ak^b\rangle_{\mathrm{ren}}(p) \ge \frac{1}{2\pi\sqrt{h(p)}} \frac{\delta^2S_{\mathrm{out}}} {\delta\lambda(p)^2} }$$

in units with $\hbar=1$. Equivalently, deform a small transverse cell of area $\mathcal A$ uniformly and take $\mathcal A\to0$; the right side is $\lim_{\mathcal A\to0}S_{\mathrm{out}}''/(2\pi\mathcal A)$. Bousso and collaborators define the diagonal derivative and normalization in Bousso et al. 2016, § 2, Eqs. (2.1)–(2.3), PDF.

The affine normalization is shared by both sides. The state, algebra or factorization regulator, choice of side, stress renormalization, entropy subtraction, and differentiability of the shape functional must be fixed. In the original local formulation, the null expansion and shear vanish at $p$; the proof below assumes the stronger condition that $\Sigma$ lies on a stationary null hypersurface.

Original stationary-surface theorem

Theorem. For free, massive or massless bosonic fields, and for the superrenormalizable bosonic extensions whose null-surface algebra and $T_{kk}$ receive no derivative-interaction counterterms, QNEC holds for arbitrary states at points on a stationary null hypersurface. In dimensions above two, the proof uses a transverse continuum limit; the free scalar result in two dimensions follows by dimensional reduction.

This is the result proved in Bousso et al. 2016, §§ 2–5, especially Eq. (4.63), PDF. “Stationary” here supplies null quantization and an everywhere nonexpanding surface. The theorem does not cover a generic dynamical cut, arbitrary marginal interactions, or fluctuating gravity.

The proof has a precise architecture:

1. Pencils. Discretize the stationary null surface into generators with small transverse area $\mathcal A$. The distinguished pencil through $p$ is a chiral $1+1$-dimensional field theory; every other degree of freedom is an auxiliary system.

2. Small-area state expansion. As $\mathcal A\to0$, the pencil is close to its vacuum. The leading off-diagonal perturbation of its density matrix scales as $\mathcal A^{1/2}$, while the auxiliary system may be arbitrary and entangled with it.

3. First-law term. Expand the entropy as $S^{(0)}+S^{(1)}+S^{(2)}+\cdots$. Vacuum null-translation invariance makes $S^{(0)}$ independent of the cut, and the vacuum modular Hamiltonian gives

   $$\left(S^{(0)}+S^{(1)}\right)'' =2\pi\mathcal A\,\langle T_{kk}\rangle.$$

4. Reduce the theorem to a sign. Terms beyond quadratic vanish faster than $\mathcal A$. QNEC is therefore equivalent to $S^{(2)\prime\prime}\le0$ at the surviving order.

5. Replica calculation. The second-order entropy is written in pencil–auxiliary modes. Hermiticity makes the relevant factors complex conjugates, and the spectral coefficient has a fixed sign, producing a negative sum of absolute squares.

This proves the inequality without assuming the null energy condition. It also explains why generic quantum-information inequalities alone are insufficient: the local null algebra, stress tensor, and vacuum modular Hamiltonian supply essential QFT structure.

General interacting flat-space result

A later modular-flow proof establishes QNEC for broad relativistic QFTs in Minkowski space, including interacting theories with the required ultraviolet and operator-product structure. It studies nested regions with locally stationary entangling surfaces, uses Tomita–Takesaki modular flow, a lightcone/defect operator expansion, analyticity in a modular-time strip, and a causality sum rule. The theorem is presented and derived in Balakrishnan et al. 2019, §§ 1–2 and §§ 5–7, PDF.

This result removes the free-field restriction in its flat-space domain; it does not automatically establish a regulator-independent inequality on every curved background. Its technical assumptions include the existence and analyticity of the modular objects and lightcone limits used in the proof. Mixed states are treated through purification. Local geometric entropy terms require a separate analysis.

Curved-space qualifications

On a curved background, both $\langle T_{kk}\rangle_{\mathrm{ren}}$ and the entropy shape derivative can shift under finite local counterterms. The physically meaningful question is whether their QNEC combination is finite and scheme independent for the dimension and congruence under study.

Fu, Koeller, and Marolf obtain the following dimension-sensitive picture in Fu, Koeller, and Marolf 2017, §§ 2–4, PDF. In the generic curved-space rows, their setup assumes a unitary ultraviolet conformal fixed point, excludes scalars saturating the unitarity bound, and restricts the background by a null convergence condition. A dot denotes affine differentiation along the selected generator.

Setting	Conditions beyond a state and cut	Conclusion
Bifurcate Killing horizon	Deformation along the horizon generator	Scheme-independent QNEC in the analyzed QFT/holographic settings
Curved spacetime, $d\le3$	$\theta=\dot\theta=0$ and $\sigma_{ab}=0$ at the point, with the source’s UV and null-convergence assumptions	Finite, scheme-independent combination and a holographic proof in the analyzed class
Curved spacetime, $d=4,5$	Sufficient vanishing derivatives through third order of expansion and shear along the null hypersurface, plus the Ricci-tensor dominant-energy restriction in Eq. (2.16)	Scheme independence and holographic proof only in that restricted class; fewer shear conditions suffice in $d=4$
Curved spacetime, $d\ge6$	Even weakly isolated horizons can allow higher-derivative counterterms	Naive renormalized QNEC is generally scheme dependent

The additional conditions are sufficient, not claimed necessary: their precise derivative form is Fu, Koeller, and Marolf 2017, § 2.2.2, Eqs. (2.15)–(2.16), PDF. In particular, imposing only $\theta=\sigma=0$ at one point is not a general curved-space theorem. The broad interacting flat-space theorem and the restricted curved-space statements remain separate. In dynamical semiclassical gravity, entropy and area counterterms combine naturally in generalized entropy; its focusing proposal is QFC, not a consequence of the flat-space QNEC proof.

Checks, counterexample, and scope

For the Minkowski vacuum and a planar Rindler cut, $\langle T_{kk}\rangle=0$. The local entropy statement needs more than invariance under a uniform translation of the whole plane: that invariance would constrain the integral of the diagonal and off-diagonal shape kernels together, allowing cancellations between them. Instead, null-plane vacuum factorization makes each transverse pencil independent, and the vacuum entropy of the half-pencil is invariant when that pencil’s endpoint is translated. Hence the diagonal $S_{\mathrm{out}}''$ vanishes and QNEC saturates. This is the local construction in Bousso et al. 2016, § 3.1 and § 3.3, Eqs. (3.14) and (3.18), PDF. Under an affine rescaling, both sides transform identically, providing an independent normalization check.

A discriminating counterexample to an unrestricted curved formulation comes from $d\ge6$: adding an allowed finite local term such as an $(\nabla R)^2$ counterterm shifts the stress and entropy derivatives so that their naive QNEC combination need not be invariant, even on a weakly isolated horizon. The state has not changed, but the claimed inequality can change scheme. This does not refute the proved stationary-surface or flat-space theorems; it shows that their hypotheses cannot be dropped.

QNEC is a fixed-background QFT statement in the proved regimes. It bounds one local null component and one diagonal entropy variation. It does not bound off-diagonal shape derivatives, prove full quantum focusing, or control metric fluctuations. Entropy must be defined by the same regulator and renormalization prescription used to establish finiteness of the QNEC combination.

Exercise

Under the positive affine reparametrization $\lambda'=a\lambda+b$, show that the two sides of QNEC transform with the same weight. Then check the planar Minkowski vacuum.

Solution

The tangent becomes $k'^a=k^a/a$, so

$$\langle T_{ab}k'^ak'^b\rangle =a^{-2}\langle T_{ab}k^ak^b\rangle.$$

The shape derivative transforms as $\delta/\delta\lambda'=a^{-1}\delta/\delta\lambda$, while the transverse metric determinant is unchanged. Hence

$$\frac{1}{2\pi\sqrt h} \frac{\delta^2S_{\mathrm{out}}}{\delta\lambda'^2} =a^{-2} \frac{1}{2\pi\sqrt h} \frac{\delta^2S_{\mathrm{out}}}{\delta\lambda^2}.$$

The inequality is affine covariant. For the Minkowski vacuum on a null plane, the renormalized stress tensor vanishes. A uniform translation of the full cut is not by itself a check of the diagonal derivative, because it also samples the off-diagonal kernel. Discretize the transverse plane instead and vary only one cell. Vacuum factorization across null pencils reduces that cell to a vacuum half-pencil, whose entropy is independent of its endpoint by null-translation invariance. Its local second derivative is therefore zero. Both sides of QNEC vanish, so the bound is saturated.

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.
• Fu, Z., Koeller, J., and Marolf, D. 2017. “The Quantum Null Energy Condition in Curved Space.” Classical and Quantum Gravity 34, 225012. DOI. Open PDF.