Quantum Null Energy Condition

The quantum null energy condition (QNEC) bounds a local null component of the renormalized stress tensor by a second null shape variation of entropy. It is a local inequality tied to an entangling cut, not ANEC and not a statement that entropy is everywhere concave.

Required background. Shape dependence and variations define the entropy derivative, and stress-tensor response kernels control the contact and response terms.

Helpful background. ANEC supplies the distinct complete-null-line inequality, while relative-entropy bounds supply the modular positivity mechanism.

Null deformation and normalization

Choose a codimension-two entangling surface and deform it along one future-directed null normal $k^\mu$ by a profile $V(y)$, where $y$ labels transverse generators. In the diagonal, local part of the second variation, QNEC takes the standard form

$$2\pi\,\langle T_{\mu\nu}k^\mu k^\nu\rangle \ge \frac{1}{\sqrt h}\, \frac{\delta^2 S_{\rm out}}{\delta V(y)^2},$$

with $\hbar=1$ and a compatible affine normalization. In two dimensions, for a cut coordinate $\lambda$, this is often written $2\pi\langle T_{kk}\rangle\ge S''(\lambda)$, with theory- and state-dependent refinements when transverse structure is absent.

Both sides scale quadratically under $k^\mu\mapsto a k^\mu$ when the deformation parameter is transformed consistently. A numerical test that rescales $T_{kk}$ but not the entropy derivative is meaningless.

General proofs use modular theory and null quantization under stated continuum assumptions; see Balakrishnan et al. 2019, §§ 2–5. Earlier proofs covered important classes of theories and states, including the formulation in Bousso et al. 2016, §§ II–IV.

Two-dimensional check

For an interval endpoint moved along a null direction in a two-dimensional CFT, compute $S(\lambda)$ and the transformed stress tensor in the same state. Form

$$\mathcal Q(\lambda) =2\pi\langle T_{kk}(\lambda)\rangle-S''(\lambda).$$

The claimed regime requires $\mathcal Q\ge0$. Test the vacuum first, then a conformally transformed state. Retain the cutoff while differentiating, subtract the state-independent divergent term, and only then take the continuum limit. A finite-difference estimate needs step-size convergence because a second derivative amplifies noise.

QNEC combines a local null stress tensor with a second shape derivative of entropy. Neither side can be replaced by the complete null integral that appears in ANEC. The diagram is schematic.

Regulator and state checks

The absolute entropy is ultraviolet divergent, while the QNEC combination is defined through a controlled renormalized variation. State-dependent surface divergences, gauge-theory edge conventions, defects, and non-smooth cuts require separate treatment. QNEC also does not license a finite-cutoff lattice inequality without demonstrating how the continuum stress tensor and entropy derivative emerge.

Null-parameter normalization, cutoff subtraction, cut smoothness, and the separation of diagonal from off-diagonal shape response are part of the theorem statement. A raw lattice second difference is not automatically QNEC. The map is schematic.

Common pitfalls

Confusing QNEC with ANEC. QNEC is local and includes entropy response; ANEC integrates stress energy over a complete generator. One may help prove the other only with additional integration and boundary control.

Differentiating after an inconsistent subtraction. Use the same geometric regulator for neighboring cuts. Otherwise cutoff motion can masquerade as $S''$.

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.
• Bousso, Raphael, Zachary Fisher, Jason Koeller, Stefan Leichenauer, and Aron C. Wall. “Proof of the Quantum Null Energy Condition.” Physical Review D 93 (2016): 024017. DOI.