Quantum Null Energy Condition (QNEC)

• Quantum Null Energy Condition (QNEC) is a rigorous inequality that bounds the local null component of the stress tensor using the second variation of entanglement entropy.
• It is derived through holographic methods, modular Hamiltonian techniques, and explicit shape derivatives in QFT, validated in both flat and curved spacetimes.
• QNEC underpins studies in quantum gravity and thermodynamics, constraining negative energy distributions and offering insights into quantum singularity theorems.

The Quantum Null Energy Condition (QNEC) is a quantum information–theoretic generalization of the classical Null Energy Condition, establishing a precise local lower bound on the expectation value of the null-null component of the stress tensor in terms of quantum entanglement entropy variations. QNEC plays a pivotal role in quantum field theory (QFT), quantum thermodynamics, and quantum gravity, particularly in characterizing the interplay between local energy densities and the structure of quantum entanglement. Its most robust proofs and applications are found in the context of holography, free field theory, and curved spacetimes.

1. Mathematical Formulation and Variants

Given a $d$-dimensional QFT on Minkowski space, let $\Sigma$ be a smooth codimension-2 surface passing through a point $p$, with $k^i$ a future-directed null vector orthogonal to $\Sigma$ at $p$. For region $R$ to one side of $\Sigma$, denote its von Neumann entropy as $S[\Sigma]$. Under an infinitesimal local null deformation parameterized by affine parameter $\lambda$ along $k^i$, the "diagonal" second variation $S''(p)$ is defined as the coefficient of the delta function in the second functional derivative of $S[\Sigma]$ in the $k^i$-direction,

$$\delta^2 S / (\delta X^i(y)\delta X^j(y'))\, k^i k^j = S''(y)\, \delta(y-y') + \text{(off-diagonal)}.$$

The QNEC states: $$\langle T_{kk}(p) \rangle \geq \frac{1}{2\pi\sqrt{h}}\, S''(p)$$ where $T_{kk} = \langle T_{ij} \rangle k^i k^j$ and $\sqrt{h}$ is the determinant of the induced metric on $\Sigma$ at $p$ (Koeller et al., 2015).

In two dimensions, QNEC is refined by an additional quadratic term in the first derivative of entropy: $$T_{kk} \geq \frac{1}{2\pi}\left[S'' + \frac{6}{c}(S')^2\right],$$ with $c$ the UV central charge, $S' = k^i \delta S/\delta X^i$ (Koeller et al., 2015, Kibe et al., 2021).

This inequality remains nontrivial in general curved backgrounds. In the most general case, a "bare" (unrenormalized) QNEC can be defined relating unrenormalized stress tensor and entropy variations, with careful attention to regularization and counterterm structure (1711.02330).

2. Holographic Derivation and General Proof Structure

Holographic QNEC

In holographic large-$N$ CFTs with classical Einstein gravity AdS duals, the entropy $S[\Sigma]$ is computed by the Ryu–Takayanagi/HRT formula,

$S[\Sigma] = \frac{A(m)}{4G_N \hbar},$

where $A(m)$ is the area of the bulk codimension-2 extremal surface $m$ homologous to $R$. For entangling cuts $\Sigma$ that are locally stationary in the null direction (i.e., $k^i K_{iab}=0$ near $p$), a key bulk causal property (Wall's achronality theorem) ensures that the union of extremal surfaces is spacelike. Expanding the bulk extremal embedding in Fefferman–Graham coordinates, the variation of bulk area yields a boundary expression for $S''$ whose finite term is proportional to $V^i(y)$, whose coefficient is related to the boundary stress tensor $T_{kk}$ through the asymptotic Einstein equation.

The resulting bound is: $$\langle T_{kk}(p) \rangle \geq \frac{1}{2\pi\sqrt{h}}\, S''(p)$$ and for $d=2$,

$$T_{kk} \geq \frac{1}{2\pi}\bigl[S'' + (6/c)(S')^2\bigr],$$

both recovered directly from the dual gravity calculations, and authoritative for classical bulk geometries (Koeller et al., 2015).

General Proofs in QFT

For general interacting QFTs, the proof of QNEC leverages the modular Hamiltonian and the properties of relative entropy under null shape deformations. Utilizing Tomita-Takesaki theory and the modular inclusion structure of the algebras associated with nested regions, one relates the second derivative of the relative entropy under null deformations to variations of the expectation value of the local stress tensor. For free fields, null quantization and the "pencil decomposition" facilitate explicit computations, with the entropy expansion revealing that the second-order correction is always nonpositive, thus proving the inequality (Balakrishnan et al., 2017, Bousso et al., 2015, Malik et al., 2019).

A simplified and general proof based on explicit shape derivatives of relative entropy was given later, bypassing the analytic continuation arguments required by earlier approaches and providing a manifestly local operator expression for the QNEC flux (Hollands et al., 6 Mar 2025).

3. Extensions: Curved Space, Bare QNEC, and Generalizations

QNEC in Curved Spacetime

In curved backgrounds, ultraviolet divergences can cause scheme dependence in both $T_{kk}$ and $S''$. It is proven that if the null congruence's expansion $\theta$ and shear $\sigma_{ab}$ vanish (and required derivatives thereof in $d=4,5$), the subtractions cancel, and QNEC remains scheme-independent and finite (Fu et al., 2017). For $d\ge 6$, certain derivative-of-curvature counterterms violate QNEC's scheme independence.

Bare QNEC

When such cancellation fails, a bare version holds for unrenormalized quantities under a physical cutoff, typically for regulated QFTs or in the presence of a UV completion (1711.02330). Here, the QNEC provides a robust off-shell energy–entropy inequality and subsumes the quantum focusing conjecture as its on-shell (semiclassical gravity) limit: $$T_{kk}^{\text{bare}}(x) \geq \frac{\hbar}{2\pi} S''_{\text{bare}}(x).$$

Stronger and Localized Inequalities

QNEC is extended to include local and integrated forms ("QNEIs"), which provide lower bounds on null energy averaged over finite intervals in both two and higher dimensions. These semi-local inequalities are derived directly from QNEC and strong subadditivity, and are universal for all interacting QFTs with a twist gap, providing the first such state-independent QNEIs beyond free fields (Fliss et al., 30 Oct 2025).

4. Special Features in Two Dimensions and the Role of Bulk Matter

In $d=2$, QNEC possesses a conformal improvement and transforms as a Virasoro primary under local conformal mappings. In pure gravity backgrounds (vacuum Bañados geometries), the QNEC is exactly saturated for all states. When the RT surface intersects nontrivial bulk matter, QNEC is generically not saturated and the gap to saturation can be computed explicitly, depending on the backreaction or quantum corrections (e.g., for a half-interval and a CFT primary of weight $h$, the gap is $h/4$ at large $h$) (Ecker et al., 2019, Stanzer, 2020).

In dynamical scenarios such as global or local quenches, QNEC constrains the rate of entanglement growth, the minimum irreversible entropy production, and the maximum possible thermalization slope after a quench. For example, QNEC bounds the increase in entropy after a global quench in holographic CFTs: $$\Delta S_\mathrm{irr} \geq \frac{\pi c}{3} \ln \left(\frac{T_f}{T_i}\right), \quad \Delta S_\mathrm{irr} \leq \frac{\pi c}{3} (T_f - T_i) L,$$ placing both lower and upper bounds dictated by quantum thermodynamics (Kibe et al., 2021, Kibe et al., 21 Mar 2025).

A further refinement, the "primary QNEC", is nontrivial in two-dimensional quenched systems and imposes strict constraints on the possible boundary states and allowed 4-point functions in conformal boundary state quenches (Kibe et al., 21 Mar 2025).

5. Saturation, Violation, and Physical Implications

Saturation and Non-Saturation

QNEC is generically saturated in vacuum or thermally equilibrium holographic states for half-space cuts and in large $d$ classical limits. For global quenches or rapidly changing geometries, exact or "fractional" saturation is observed: in two-dimensional AdS-Vaidya global quench, the large-interval QNEC saturates only halfway (i.e., QNEC is only "half-saturated") (Ecker et al., 2019). For far-from-equilibrium colliding shockwave states, QNEC can be saturated in one null direction while classical energy conditions are violated in the other, establishing that QNEC is both stronger and weaker than the NEC in different contexts (Ecker et al., 2017).

Violation and Limits

The first explicit violation of QNEC was found in a strongly coupled holographic CFT on a wormhole background where large infrared (IR) entanglement contributions, not present in conventional equilibrium or confining phases, dominate and invalidate the bound. This shows that QNEC as usually formulated requires careful subtraction of IR-divergent contributions and is not unconditionally valid for all states or geometries; local (UV) QNEC, however, can remain valid (Ishibashi et al., 2018).

Quantum Information Interpretation

QNEC governs the possible distributions of negative energy densities in QFT, plays a central role in quantum singularity theorems, and provides state-by-state bounds on entanglement dynamics and entropy production. Its structural link to relative entropy, strong subadditivity, and modular theory connects it deeply to quantum information theory. The QNEC can be equivalently formulated as the positivity of the second null-shape variation of relative entropy, and generalizes to a "Rényi QNEC," for sandwiched Rényi divergences of order $n>1$ in free fields (Moosa et al., 2020, Roy, 2022).

6. Table: QNEC—Key Forms, Regimes, and Status

Context	Leading QNEC Formulation	Status/Features
Minkowski, $d\ge 3$	$\langle T_{kk}\rangle \geq (1/2\pi) S''$	Proven for free, holographic, and interacting QFTs
Two dimensions ($d=2$)	$T_{kk} \geq (1/2\pi) [S'' + (6/c)(S')^2]$	Saturated in pure gravity; gap in presence of matter
Curved space	$T_{kk}\geq (1/2\pi) S''$, under expansion/shear vanishing	Scheme independent for appropriate cuts and $d\le5$
Bare QNEC	$$T_{kk}^{\text{bare}} \geq (\hbar/2\pi) S''_{\text{bare}}$$	General curved backgrounds (regulated), all $d\le4$, smeared in $d=5$
Holographic CFT (large $N$)	$\langle T_{kk}\rangle \geq (1/2\pi \sqrt{h}) S''$	Leading order in $1/N$, classical Einstein bulk
Global/local quenches ($d=2$)	Bounds on entropy growth, e.g., $$\Delta S_{\rm irr} \geq \frac{\pi c}{3}\ln\frac{T_f}{T_i}$$	Lower and upper bounds on entropy/temperature change

7. Applications, Impact, and Future Directions

QNEC underlies modern formulations of "quantum energy conditions," replacing the role of classical energy conditions in semi-classical gravity, quantum singularity theorems, and generalized second laws. It provides robust, local, and quantitative bounds in both equilibrium and far-from-equilibrium quantum dynamics, critical for strongly coupled systems and quantum thermodynamics (Kibe et al., 2021). The interplay of QNEC with modular theory, relative entropy, and entropy currents has fueled new developments in entropy inequalities, quantum gravity/no-go theorems, and quantum memory protection (Hollands et al., 6 Mar 2025, Banerjee et al., 2022).

Current open directions include deriving fully local QNEIs in higher dimensions, generalizing QNEC to Renyi and other divergences in interacting QFTs, refining its status in spacetimes with nontrivial infrared structure, and coupling with gravity to further constrain semiclassical spacetime geometries (Fliss et al., 30 Oct 2025, Roy, 2022).

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References:

(Koeller et al., 2015) "Holographic Proof of the Quantum Null Energy Condition" (Kibe et al., 2021) "Quantum thermodynamics of holographic quenches and bounds on the growth of entanglement from the QNEC" (1711.02330) "Bare Quantum Null Energy Condition" (Fu et al., 2017) "The Quantum Null Energy Condition in Curved Space" (Kibe et al., 21 Mar 2025) "Quantum null energy condition in quenched 2d CFTs" (Ecker et al., 2019, Stanzer, 2020, Ecker et al., 2017, Moosa et al., 2020, Roy, 2022, Ishibashi et al., 2018, Bousso et al., 2015, Hollands et al., 6 Mar 2025, Fliss et al., 30 Oct 2025, Khandker et al., 2018, Fu et al., 2016).