Quantum Null Energy Condition

The quantum null energy condition (QNEC) says that the renormalized null energy at a point cannot fall below a specific local second shape variation of entanglement entropy. The word “shape” is essential: one moves a small part of an entangling cut along an affinely parametrized null generator. QNEC is neither a pointwise positivity condition on energy nor a synonym for a null average.

Required background. Shape dependence and variations defines functional entropy derivatives. Stress-tensor response kernels supplies the distinction between contact and separated-point response.

Helpful background. ANEC gives the different complete-null-line statement. Relative-entropy bounds provides a regional comparison whose modular methods also enter modern QNEC proofs.

For the chapter-wide orientation, use the entry map, the comparison of claim domains, and the controls that change conclusions.

The diagonal null shape derivative

Work first in $d$-dimensional Minkowski spacetime. Let $\Sigma$ be a smooth codimension-two surface that divides a Cauchy slice, and let $k^\mu(y)$ be a future-directed null normal near a point $p\in\Sigma$; $y$ labels the transverse generators. Choose an affine parameter and deform the embedding by

$$X^\mu(y;V)=X_0^\mu(y)+V(y)k^\mu(y)$$

on a locally stationary null hypersurface. Write the entropy of one chosen side as the functional $S[V]$. A convention that exposes the density factors is

$$\delta^2 S =\int d^{d-2}y\sqrt h\int d^{d-2}y'\sqrt{h'}\, \delta V(y)\delta V(y') \left[S_{\mathrm{diag}}''(y)\,\delta_h(y,y') +S_{\mathrm{off}}''(y,y')\right],$$

where $\delta_h$ is the invariant transverse delta function,

$$\int d^{d-2}y'\sqrt{h'}\,\delta_h(y,y')f(y')=f(y).$$

QNEC uses only the diagonal coefficient:

$$\boxed{2\pi\,\langle T_{\mu\nu}(p)\rangle k^\mu k^\nu \geq S_{\mathrm{diag}}''(p)} \qquad(\hbar=1).$$

If instead one defines a coordinate functional derivative without invariant measures, the same statement is commonly displayed as $2\pi\langle T_{kk}\rangle\geq S''/\sqrt h$. The two forms agree only after the delta-function and measure conventions are matched. A finite deformation of an extended patch contains the off-diagonal kernel as well and is not the local quantity in the box.

The first proof applied to free and superrenormalizable bosonic theories on stationary null surfaces Bousso et al. 2016, §§ II–V. Holographic proofs treated large-$N$ theories with suitable Einstein-gravity duals Koeller and Leichenauer 2016, §§ 2–3. The general modular-flow proof establishes the result for relativistic continuum QFTs under its light-cone, operator-product, state-domain, and cut-regularity assumptions Balakrishnan et al. 2019, §§ 2 and 6–7. These are theorem hypotheses, not interchangeable labels for an arbitrary lattice model.

Affine normalization is part of the observable

Let $\lambda$ be affine and $k^\mu=dX^\mu/d\lambda$. Under a constant rescaling $\widetilde\lambda=a\lambda$,

$$\widetilde k^\mu=\frac{k^\mu}{a}, \qquad T_{\widetilde k\widetilde k}=\frac{T_{kk}}{a^2}, \qquad \frac{d^2S}{d\widetilde\lambda^2} =\frac1{a^2}\frac{d^2S}{d\lambda^2}.$$

QNEC is therefore covariant under affine rescaling. Equivalently, if one sends $k\mapsto ak$ while keeping the deformation label fixed, both sides scale as $a^2$.

A nonlinear reparameterization is different. If $\widetilde\lambda=f(\lambda)$, then

$$\frac{d^2S}{d\widetilde\lambda^2} =\left(\frac{d\lambda}{d\widetilde\lambda}\right)^2S'' +\frac{d^2\lambda}{d\widetilde\lambda^2}S'.$$

The stress tensor has no term proportional to $S'$. Thus the elementary QNEC formula presupposes affine parametrization; a nonlinear relabeling cannot be used as a harmless numerical convenience.

Renormalization and geometric hypotheses

The entropy of a sharp region is ultraviolet divergent. In the standard flat-space statement, the cut is smooth, the null deformation is evaluated locally on a stationary null surface, and the stress tensor and entropy variation use one compatible renormalization prescription. Local geometric counterterms that could contaminate the diagonal second variation are absent or controlled under these hypotheses. The condition is often written

$$k_iK^i_{ab}=0$$

at the point, with $K^i_{ab}$ the extrinsic curvature of the cut. Koeller and Leichenauer 2016, §2 carefully separates the diagonal term and explains the stationary-cut condition.

This does not automatically establish the same formula for a corner, a boundary defect, a curved background, a finite lattice cutoff, or a gauge-theory factorization with unspecified edge data. Such cases can possess additional local terms or require a theorem adapted to their algebra. A reliable computation deforms the regulated cut first, keeps the regulator geometrically matched for neighboring shapes, identifies the diagonal distributional coefficient, and only then takes the continuum limit.

Benchmark: vacuum and thermal saturation in a 2D CFT

Two dimensions have no transverse coordinate, and a CFT admits the stronger conformally covariant form

$$2\pi\langle T_{++}\rangle \geq S''+\frac6c(S')^2.$$

Use null coordinates $x^\pm=t\pm x$ with $ds^2=dx^+dx^-$ in the site’s (+---) convention, and vary one interval endpoint in the affine coordinate $\lambda=x^+$. Keep the other null separation fixed. The derivative is then with respect to $L^+=x_2^+-x_1^+$, and $c$ is the CFT central charge.

For the vacuum, the varying chiral part of the entropy is

$$S_+(L^+)=\frac c6\log\!\left(\frac{L^+}{\epsilon}\right).$$

Hence $S_+'=c/(6L^+)$ and $S_+''=-c/[6(L^+)^2]$, so

$$S_+''+\frac6c(S_+')^2=0=2\pi\langle T_{++}\rangle_{0}.$$

The cutoff cancels from the derivatives, but that cancellation relies on holding the same affine-coordinate cutoff convention while moving the endpoint.

Now take a thermal state of inverse temperature $\beta$. The interval formula of Calabrese and Cardy 2004, §3 gives

$$S_+(L^+) =\frac c6\log\!\left[ \frac{\beta}{\pi\epsilon} \sinh\!\left(\frac{\pi L^+}{\beta}\right) \right].$$

With $a=\pi/\beta$,

$$S_+'=\frac{ca}{6}\coth(aL^+), \qquad S_+''=-\frac{ca^2}{6}\operatorname{csch}^2(aL^+).$$

Using $\coth^2u-\operatorname{csch}^2u=1$,

$$S_+''+\frac6c(S_+')^2=\frac{ca^2}{6}.$$

The thermal stress tensor is $\langle T_{++}\rangle=\pi c/(12\beta^2)$, so $2\pi\langle T_{++}\rangle=ca^2/6$: the stronger two-dimensional QNEC saturates. For $c=1$, $\beta=2\pi$, and $L^+=2$,

$$S_+'=0.10941961, \qquad S_+''=-0.03016924, \qquad S_+''+6(S_+')^2=0.04166667,$$

while $T_{++}=0.006631456$ and $2\pi T_{++}=0.04166667$. Omitting the nonlinear term would produce a true but non-sharp test and would miss conformal covariance in this example.

QNEC, QEI, and ANEC answer different questions

Statement	Geometry	Quantity constrained	Essential extra data
QNEC	One point on a null-deformed cut	$T_{kk}$ versus a diagonal entropy shape derivative	Cut, side, affine normalization, renormalization
ANEC	A complete affine null geodesic	Integral of $T_{kk}$ along the generator	Completeness and endpoint behavior
QEI	Usually a timelike worldline with smooth sampling	Weighted average of renormalized energy density	Field, state class, trajectory, sampler

Integrating a local QNEC expression does not become ANEC unless the entropy-derivative boundary terms and the relation between local and off-diagonal variations are controlled. Conversely, ANEC contains no local entropy response. QEI bounds can quantify negative-energy duration but generally involve a different trajectory and sampling functional.

Failure controls

Wrong derivative. A spatial second derivative, a total interval-size derivative, and the diagonal null functional derivative are different objects.

Wrong parameter. Rescale the affine parameter consistently as a quick covariance check. A nonlinear relabeling introduces an $S'$ term and changes the formula.

Wrong regulator. If the cutoff surface moves differently for neighboring cuts, a local counterterm can masquerade as physical $S''$.

Wrong theorem domain. Boundaries, gauge edges, curved backgrounds, singular cuts, or finite lattice spacing require their own control. Zero numerical QNEC gap at one resolution is not a continuum theorem.

Wrong two-dimensional formula. In a 2D CFT, test the conformally covariant combination with $6(S')^2/c$ when claiming the stronger result.

Exercises

1. Check affine-rescaling covariance.

Solution

For $\widetilde\lambda=a\lambda$, the tangent is $\widetilde k=k/a$, so $T_{\widetilde k\widetilde k}=T_{kk}/a^2$. The chain rule gives $d^2S/d\widetilde\lambda^2=S''/a^2$ because $a$ is constant. Both sides acquire the same factor.

2. Verify vacuum saturation of the stronger 2D expression.

Solution

From $S_+=(c/6)\log(L^+/\epsilon)$, one obtains $S_+'=c/(6L^+)$ and $S_+''=-c/[6(L^+)^2]$. The nonlinear term equals $c/[6(L^+)^2]$, so the sum vanishes, matching the vacuum value of $2\pi T_{++}$.

3. Reproduce the thermal benchmark analytically.

Solution

Differentiate the logarithm of $\sinh(aL^+)$. The identity $\coth^2u-\operatorname{csch}^2u=1$ reduces the entropy combination to $ca^2/6$. Since $a=\pi/\beta$ and $T_{++}=\pi c/(12\beta^2)$, the stress side is also $ca^2/6$.

4. Explain why a nonlinear parameter change is not an affine normalization change.

Solution

The second derivative gains $(d^2\lambda/d\widetilde\lambda^2)S'$. The transformed null tangent only rescales $T_{kk}$ by $(d\lambda/d\widetilde\lambda)^2$, so there is no stress-tensor term that can match the extra contribution. The standard formula therefore applies only after fixing an affine parameter.

References

• Balakrishnan, Souvik, Thomas Faulkner, Zuhair U. Khandker, and Huajia Wang. “A General Proof of the Quantum Null Energy Condition.” Journal of High Energy Physics 2019, no. 9 (2019): 020. DOI. Open paper.
• Bousso, Raphael, Zachary Fisher, Jason Koeller, Stefan Leichenauer, and Aron C. Wall. “Proof of the Quantum Null Energy Condition.” Physical Review D 93, no. 2 (2016): 024017. DOI. Open paper.
• Calabrese, Pasquale, and John Cardy. “Entanglement Entropy and Quantum Field Theory.” Journal of Statistical Mechanics: Theory and Experiment 2004, no. 6 (2004): P06002. DOI. Open paper.
• Koeller, Jason, and Stefan Leichenauer. “Holographic Proof of the Quantum Null Energy Condition.” Physical Review D 94, no. 2 (2016): 024026. DOI. Open paper.