Quantum Null Energy Inequalities

• Quantum Null Energy Inequalities (QNEIs) are state-independent lower bounds on smeared null energy integrals in quantum field theory, ensuring physically consistent energy conditions.
• They leverage methods from QNEC and timelike quantum inequalities to control null energy flux along boosted timelike curves and curved-spacetime geodesics.
• These bounds play a key role in excluding exotic phenomena like wormholes and time machines by constraining accumulations of negative energy.

Searching arXiv for recent and foundational work on Quantum Null Energy Inequalities and closely related null-energy bounds. Searching specifically for papers deriving QNEIs from QNEC and for classic ANEC-from-QI curved-spacetime results. Quantum Null Energy Inequalities (QNEIs) are state-independent lower bounds on suitably smeared or integrated null energy in quantum field theory. In the modern literature, the subject includes both direct lower bounds on null-energy flux, typically written in terms of $T_{vv}$, and closely related “null-projected quantum inequalities” that average $T_{ab}\ell^a\ell^b$ along highly boosted timelike worldlines and then use the result to control null averages and prove achronal ANEC. A neighboring but distinct development is the Quantum Null Energy Condition (QNEC), which gives a local entropy-corrected lower bound on null energy rather than a standard smeared inequality. The resulting landscape is technically heterogeneous: some results are explicit QNEIs, some are ANEC-from-QI theorems, and some are entropic null-energy bounds that constrain or generate QNEIs indirectly (Fliss et al., 30 Oct 2025, Kontou et al., 2012, 1711.02330).

1. Conceptual scope and taxonomy

The literature distinguishes several null-energy statements that are related but not interchangeable. A QNEI is a lower bound on a smeared null energy functional, while ANEC is a complete null average, and QNEC is a pointwise lower bound involving entropy derivatives rather than a sampling function (Fliss et al., 30 Oct 2025, Kontou et al., 2012, 1711.02330).

Notion	Representative expression	Role in the literature
QNEI	$$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$	State-independent lower bound on smeared null energy
ANEC	$$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$	Complete null average relevant to topological censorship and chronology protection
QNEC	$T_{kk}-\frac{1}{2\pi}S'' \ge 0$	Local entropy-corrected null-energy bound
Timelike QEI	$\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$	Worldline-averaged lower bound, often the technical input for null results

This taxonomy matters because many papers commonly grouped with QNEIs do not actually prove a direct finite null-smearing inequality. The curved-spacetime achronal-ANEC papers derive an ANEC theorem from a timelike-averaged inequality for the null projection $T_{ab}\ell^a\ell^b$, not from a direct null-worldline bound (Kontou et al., 2012). Conversely, recent work derives genuine semi-local QNEIs directly from QNEC, strong subadditivity, defect operator expansions, and modular-Hamiltonian technology (Fliss et al., 30 Oct 2025). The QNEC literature is adjacent rather than identical: it studies the local combination

$Q:=T_{ab}k^a k^b-\frac{1}{2\pi}S''\ge 0,$

or, in $2d$, the stronger form

$$2\pi \langle T_{vv}(v)\rangle \ge \partial_v^2 S(v)+\frac{6}{c}(\partial_v S(v))^2,$$

which is not itself a smeared QNEI (1711.02330, Fliss et al., 30 Oct 2025).

2. Timelike worldline technology and null projection

A large part of the subject developed from timelike quantum inequalities. The basic input used in later null applications is a flat-spacetime timelike-averaged inequality for the null contraction: $T_{ab}\ell^a\ell^b$0 Here $T_{ab}\ell^a\ell^b$1 is a timelike geodesic segment, $T_{ab}\ell^a\ell^b$2 its tangent, $T_{ab}\ell^a\ell^b$3 a constant null vector, and $T_{ab}\ell^a\ell^b$4 smooth, real, and compactly supported. The important point is that this is not a null-average inequality; it is a timelike average of the null-contracted stress tensor (Kontou et al., 2012).

The curved-spacetime extension is correspondingly indirect. One line of work formulates a conjectured curved-space version with local curvature corrections and Wald-type renormalization ambiguities. Another line derives such a bound perturbatively to first order in curvature and its derivatives for a massless minimally coupled scalar field, again along timelike paths rather than directly on null worldlines (Kontou et al., 2012, Kontou et al., 2015). In the explicit perturbative form, the lower bound retains the flat-space leading term and adds controlled curvature corrections, schematically

$T_{ab}\ell^a\ell^b$5

with $T_{ab}\ell^a\ell^b$6, $T_{ab}\ell^a\ell^b$7, and $T_{ab}\ell^a\ell^b$8 a large boost parameter (Kontou et al., 2015).

This suggests a characteristic strategy for QNEI-adjacent null results: one does not average directly along a null geodesic but instead reinterprets a long, thin null parallelogram as a family of highly boosted timelike segments. In the boost limit, the timelike inequality becomes strong enough to obstruct uniformly negative null averages. That mechanism is central to the curved-spacetime ANEC proofs and remains one of the clearest links between rigorous timelike QEIs and null-energy constraints (Kontou et al., 2012, Kontou, 2015).

3. Curved spacetime, achronal ANEC, and no-go theorems for exotic geometry

The curved-spacetime ANEC results apply to a classical background $T_{ab}\ell^a\ell^b$9 containing an achronal null geodesic $$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$0 surrounded by a tubular neighborhood $$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$1 foliated by nearby achronal null geodesics. The local geometric assumptions are strong: smooth bounded curvature and bounded derivatives, localized curvature, causal isolation of the tube, and the null convergence condition

$$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$2

for every null $$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$3. The quantum field is a free, minimally coupled scalar field in a Hadamard state, with the renormalized stress tensor defined in Wald’s framework (Kontou et al., 2012, Kontou et al., 2015).

The theorem proved in this setting is not phrased as a direct pointwise statement for a single chosen geodesic. Rather, it states that it is impossible for the ANEC integral

$$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$4

to converge uniformly to negative values on all geodesics $$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$5 in the tube. The emphasis on a finite congruence is essential because the intended application is the exclusion of macroscopic exotic geometries such as wormholes and time machines, not merely the behavior of a single isolated geodesic (Kontou et al., 2012).

The proof strategy has a standard architecture. One assumes a uniformly negative null average on the tube, integrates $$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$6 over a null parallelogram, and obtains a negative upper bound. The same set of points is then foliated by highly boosted timelike curves, to which the null-projected timelike QI is applied. The geometric work enters in showing that the boosted curves remain timelike, that their proper acceleration is small, that the relevant causal diamonds lie inside $$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$7, and that dangerous boosted curvature components are absent. Under achronality and null convergence, one derives identities such as

$$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$8

which ensure that curvature terms do not spoil the boost limit. Comparing the asymptotic scaling of the QI lower bound with the assumed negative upper bound then yields a contradiction (Kontou et al., 2012, Kontou et al., 2015).

These papers are therefore best classified as ANEC-from-QI results, not as direct finite null-sampling QNEIs. Their significance for the encyclopedia topic lies in the fact that they show how null-projected timelike inequalities can imply achronal ANEC in curved spacetime under controlled geometric hypotheses, and thereby constrain exotic causal structure (Kontou, 2015).

4. Entropy-corrected local null bounds and their relation to QNEIs

The QNEC literature studies a different object: a local lower bound on null energy in terms of entropy variations. Holographically, the foundational statement is

$$\int_D d^p x\, g(x)\,\langle T_{vv}(x)\rangle \ge -F[g]$$9

or, in equivalent local notation, $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$0. This is a pointwise bound, not a sampling-function inequality. In holographic large-$$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$1 theories it was proved at leading order in large $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$2 for CFTs and relevant deformations in Minkowski space with Einstein gravity duals (Koeller et al., 2015).

On curved backgrounds the situation is subtler. The renormalized QNEC is naturally finite and scheme-independent in $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$3 under local stationarity conditions such as $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$4 and $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$5. In $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$6 more conditions are required, including vanishing of additional derivatives and a dominant energy condition, while in $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$7 the corresponding pointwise renormalized QNEC generally fails even on weakly isolated horizons because derivative curvature counterterms can shift the combination $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$8 (Fu et al., 2017). A closely related analysis shows that, in $$\int_{-\infty}^{\infty} d\lambda\, T_{ab}\ell^a\ell^b \ge 0$$9, Entanglement Wedge Nesting implies QNEC under necessary and sufficient geometric conditions, and that the Quantum Focusing Conjecture yields the same QNEC under those same conditions (Akers et al., 2017).

When renormalized finiteness fails, one can instead formulate a bare QNEC using unrenormalized stress tensor and entropy defined with a common physical regulator. In this formulation the divergent contributions to the bare quantity $T_{kk}-\frac{1}{2\pi}S'' \ge 0$0 have the correct sign for $T_{kk}-\frac{1}{2\pi}S'' \ge 0$1, while in $T_{kk}-\frac{1}{2\pi}S'' \ge 0$2 the pointwise bare QNEC can fail but a smeared version survives in a holographic setting with Fefferman–Graham regulators (1711.02330). The same general program also admits information-theoretic refinements. A Rényi-QNEC, formulated using second null shape derivatives of sandwiched Rényi divergence, is proved for free and superrenormalizable theories in $T_{kk}-\frac{1}{2\pi}S'' \ge 0$3 when $T_{kk}-\frac{1}{2\pi}S'' \ge 0$4, while explicit counterexamples exist for $T_{kk}-\frac{1}{2\pi}S'' \ge 0$5 (Moosa et al., 2020). For free fermionic field theories, a direct proof of QNEC using null quantization and replica methods shows

$T_{kk}-\frac{1}{2\pi}S'' \ge 0$6

under the standard stationary-null-surface assumptions (Malik et al., 2019).

This body of work is not standard QNEI theory, but it heavily shapes it. It clarifies when local null-energy lower bounds are meaningful, how entropy variations compensate violations of classical NEC, and how local QNEC statements interface with integrated conditions such as ANEC. In holographic interacting theories, the diagonal part of QNEC can even be saturated, with

$T_{kk}-\frac{1}{2\pi}S'' \ge 0$7

so that null energy is identified with the local contact term in the second shape variation of entanglement entropy (Leichenauer et al., 2018).

5. Interacting theories and the limits of current control

Direct interacting-theory results remain much stronger for timelike QEIs than for QNEIs. In the massive Ising model in $T_{kk}-\frac{1}{2\pi}S'' \ge 0$8-dimensional Minkowski space, a state-independent lower bound on time-averaged energy density was proved: $T_{kk}-\frac{1}{2\pi}S'' \ge 0$9 This is the first QEI established for an interacting quantum field theory with nontrivial $\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$0-matrix, but it is not a QNEI because it concerns $\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$1 averaged along a timelike worldline rather than $\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$2 averaged along a null curve (Bostelmann et al., 2013).

A broader nonperturbative analysis of interacting $\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$3-dimensional integrable quantum field theories reaches the same conclusion. The central lower bound is of the form

$\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$4

again along timelike worldlines. The paper shows how the existence or failure of such a QEI is controlled by the one-particle form factor $\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$5, and it proves a full-Hilbert-space Ising-model result while obtaining one-particle-level results for more general scalar integrable models. The paper explicitly notes that it does not derive ANEC, QNEIs, or any lower bound for null-contracted stress-tensor averages (Cadamuro, 2019).

This separation between timelike and null results is structurally informative. It suggests that state-independent lower bounds on averaged local energy can survive genuine self-interaction, but also that extending such results to null averaging is substantially harder. The timelike literature provides concrete obstacles: stress-tensor ambiguities, stronger local negativity in interacting theories, and sensitivity to large-rapidity behavior. These features plausibly explain why direct interacting-theory QNEIs were long absent even after interacting timelike QEIs became available (Bostelmann et al., 2013, Cadamuro, 2019). Exact closed timelike-worldline QEIs along stationary worldlines reinforce this contrast: even in flat space, explicit lower bounds rely on the favorable microlocal structure of timelike pullbacks, a feature that degenerates in the null limit (Fewster et al., 2023).

6. Semi-local QNEIs from QNEC and recent directions

A recent development derives new families of genuine QNEIs directly from QNEC. In $\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$6, the full QNEC

$\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$7

yields, after integration against a nonnegative smearing function $\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$8, the Fewster–Hollands bound

$\int dt\, g(t)^2\,\langle T^{00}\rangle \ge -c_g$9

The same work then uses only the linear part of QNEC, together with strong subadditivity, monotonicity of relative entropy, and modular Hamiltonians for null intervals, to derive a broader family of $T_{ab}\ell^a\ell^b$0 QNEIs with corrected smearing $T_{ab}\ell^a\ell^b$1 depending on an auxiliary function $T_{ab}\ell^a\ell^b$2 (Fliss et al., 30 Oct 2025).

In higher dimensions the situation is more delicate. A single-null-line finite-segment QNEI of the same type is obstructed, so the derived bounds are semi-local rather than purely one-dimensional: they smear in the null direction $T_{ab}\ell^a\ell^b$3, also smear in the second null direction $T_{ab}\ell^a\ell^b$4, and integrate uniformly over transverse directions. The resulting higher-dimensional bounds apply to interacting CFTs in $T_{ab}\ell^a\ell^b$5 under a twist-gap assumption, with the stress tensor as the lowest-twist nontrivial operator in the null defect OPE. Their right-hand side is state-independent and depends only on the smearing, the stress-tensor two-point coefficient $T_{ab}\ell^a\ell^b$6, and the vacuum strip coefficient $T_{ab}\ell^a\ell^b$7; the paper describes them as the first state-independent lower bounds of this kind for interacting theories in higher dimensions (Fliss et al., 30 Oct 2025).

Recent quench studies illustrate how the QNEC side of the subject feeds back into QNEIs without becoming identical to them. In homogeneous quenches, QNEC bounds the early-time quadratic entanglement-growth coefficient by

$T_{ab}\ell^a\ell^b$8

and this bound is saturated by boundary-state quenches in $T_{ab}\ell^a\ell^b$9 CFT and in higher-dimensional holographic CFTs (Mezei et al., 2019). In quenched $Q:=T_{ab}k^a k^b-\frac{1}{2\pi}S''\ge 0,$0 CFTs, primary QNEC imposes bounds on twist-field four-point data, and violation of primary QNEC implies violation of ANEC in a conformally transformed frame (Kibe et al., 21 Mar 2025). These are not direct QNEIs, but they show that entropy-based null bounds can constrain nonequilibrium dynamics and can control the same null-energy sector that QNEIs probe.

Taken together, the literature defines QNEIs as a layered subject. At one end are direct state-independent lower bounds on smeared null energy, now available in $Q:=T_{ab}k^a k^b-\frac{1}{2\pi}S''\ge 0,$1 and in semi-local higher-dimensional interacting settings (Fliss et al., 30 Oct 2025). In the middle are null-projected timelike inequalities used to prove achronal ANEC in curved spacetime (Kontou et al., 2012, Kontou et al., 2015). Alongside them lies the QNEC program, which supplies local entropy-corrected null-energy bounds, clarifies renormalization issues in curved space, and increasingly serves as a generator of genuine QNEIs rather than merely a neighboring concept (Koeller et al., 2015, Fu et al., 2017).