Quantum Marginal and Trapped Surfaces

In the two-sided convention used for local horizon classification here, a future quantum trapped surface is a compact codimension-two cut whose two future-directed quantum expansions are negative; a future quantum marginal surface has one vanishing expansion and the other nonpositive. A quantum singularity theorem in the literature instead uses a distinct one-sided outward definition. Either convention is useful only after the null orientations, outside algebra, quantum state, renormalization scheme, and slicing are fixed. Unlike classical trapping, the classification is not a local property of the metric alone.

Required background. Quantum Information supplies reduced-state entropy and algebra choice, which make the quantum expansions state dependent and nonlocal.

Helpful background. Quantum Fields in Curved Spacetime supplies admissible states and renormalized entropy; Einstein Equations supplies the mean geometry; Black-Hole Geometry and Mechanics supplies classical trapped surfaces and null normals; and Semiclassical Gravity fixes the approximation in which these ingredients are combined.

Units. This page restores $\hbar$ in quantum-expansion formulas while keeping $c=k_{\rm B}=1$; setting $\hbar=1$ recovers the site-wide convention.

The two quantum-expansion signs

Choose independent future-directed null normals $k^a$ and $\ell^a$ to a smooth cut $\sigma$, with $k\cdot\ell=-1$. Choose which side of the cut defines $S_{\rm out}$ and use the same choice for both deformations. The two quantum expansions are

$$\Theta_k =\frac{4G\hbar}{\sqrt\gamma}\frac{\delta S_{\rm gen}}{\delta V_k}, \qquad \Theta_\ell =\frac{4G\hbar}{\sqrt\gamma}\frac{\delta S_{\rm gen}}{\delta V_\ell}.$$

With the stated future orientation, the classifications are:

Surface type	$\Theta_k$	$\Theta_\ell$	Meaning
Future quantum trapped	$\lt0$	$\lt0$	Generalized entropy initially decreases along both future null deformations
Future quantum marginal	$=0$	$\le0$	One future deformation is stationary and the other is nonincreasing
Quantum untrapped	opposite signs	opposite signs	One future null direction increases generalized entropy
Future quantum antitrapped	$\gt0$	$\gt0$	Both future deformations increase generalized entropy

The nonstrict marginal convention includes the doubly stationary case. A strictly marginally trapped cut has the second expansion negative; a quantum extremal surface has both zero. These distinctions matter in screen-evolution theorems, where a definite second sign and additional genericity assumptions are needed.

Positive rescalings $k^a\mapsto c k^a$ and $\ell^a\mapsto c^{-1}\ell^a$ preserve all these sign classes. Reversing a null orientation does not; “future” is part of the definition. In the leading gravitational limit, $\Theta_k\to\theta_k$ and $\Theta_\ell\to\theta_\ell$, so the familiar classical signs are recovered.

A distinct one-sided definition is used in Wall 2013, § 3.2, Theorems 3–4, PDF. Take a globally hyperbolic spacetime, a compact cut dividing a connected Cauchy slice into an interior and a noncompact exterior, semiclassical control near the cut, and decreasing fine-grained generalized entropy along its outward future null hypersurface at every point. Assuming the fine-grained GSL for the relevant causal horizons, the theorem implies null geodesic incompleteness. It does not require the second future quantum expansion to be negative, establish curvature blowup, or specify a microscopic endpoint. A two-sided future quantum trapped cut supplies the outward sign, but still needs all these global and GSL hypotheses.

State, scheme, and slicing dependence

The geometric expansions depend on the embedding of $\sigma$ and hence on slicing. Quantum expansions inherit that dependence and add nonlocal state information. Moving the cut near $y$ changes which degrees of freedom belong to the outside algebra; entanglement with distant degrees of freedom can therefore alter the derivative at $y$. A surface that is marginal in one state need not be marginal in another on the same mean metric.

Renormalization does not make the sign arbitrary. When the effective gravitational couplings and exterior entropy are translated together, $S_{\rm gen}$ and its shape derivative are finite. But changing the physical state, the cut, or the outside algebra changes the observable. A calculation that holds the bare area term fixed while changing only the entropy subtraction confuses scheme dependence with physical state dependence.

There is also no general slicing-independent “quantum apparent horizon” theorem. One can study quantum marginal tubes formed by stacking marginal cuts, but existence, uniqueness, signature, and stability require additional evolution and genericity hypotheses. Quantum extremal surfaces differ again: both normal variations vanish, usually with anchoring data, while a marginal surface keeps a preferred null direction and causal interpretation.

Evidence and the scope of the horizon claims

The sources checked through 2026-09-26 support the definition and several controlled implications, but not a universal horizon-existence or singularity-resolution theorem.

Evidence or formulation	Result for the signs	Scope and limitation
Generalized-entropy definition in Bousso et al. 2016, §§ 3.2–3.3, PDF	Defines finite $\Theta$ and proposes monotonicity under further null deformation	The original unrestricted pointwise proposal has counterexamples; the derivative remains state- and cut-dependent
Quantum singularity theorem in Wall 2013, § 3.2, Theorem 4, PDF	The one-sided outward decrease, noncompact exterior, global hyperbolicity, local semiclassical control, and fine-grained GSL imply null incompleteness	Its “quantum trapped” definition is not the two-expansion sign class above; it neither proves formation nor identifies a microscopic endpoint
Quantum-extremal-surface barriers in Engelhardt and Wall 2015, §§ 4–6, PDF	Nonincreasing generalized entropy can obstruct certain extremal surfaces in a perturbative holographic setting	Holographic anchoring and local bulk propagation are additional assumptions, not generic horizon data
Restricted-focusing analysis in Franken et al. 2026, §§ 3–4, PDF	Proves zero-expansion focusing in specified JT/QFT models and derives a CFT bound conditional on higher-dimensional rQFC	Published result, but neither the higher-dimensional rQFC nor generic quantum-marginal evolution is thereby proved

The conservative position treats $\Theta_k$ and $\Theta_\ell$ as well-defined semiclassical diagnostics once state, algebra, cut, and counterterms are specified, while withholding global claims. A more permissive position expects appropriate restricted focusing principles to give robust evolution and barrier laws. The two positions are not contradictory until they are compared on the same background, state class, deformation, regulator, and semiclassical order.

Counterexamples and discriminating checks

Classical-limit check. If the outside entropy is constant under the deformation and higher-curvature terms are negligible, the quantum and classical signs coincide. This verifies normalization but says nothing about states with large entropy gradients.

Near-marginality check. If $\theta_k$ is already as small as the retained entropy correction, its sign alone does not determine the sign of $\Theta_k$. The solved example below shows the algebraic possibility. Realizing specified entropy gradients in a physical state requires a separate state construction; this sign exercise does not establish that arbitrary gradients and stress tensors can be chosen independently.

Dropped-global-hypothesis check. A local patch with both quantum expansions negative is not by itself the compact surface on a noncompact Cauchy slice required by the singularity architecture. Nor does a trapping-horizon tube automatically satisfy a causal-horizon generalized second law. These failures distinguish a definition from its global consequences.

The status remains unresolved for generic interacting fields on dynamical four-dimensional backgrounds and for large metric fluctuations. Non-Hadamard states can fall outside the construction altogether. The conclusions could change after a proof or counterexample changes the domain of a restricted focusing law, a self-consistent evolution establishes or destroys a quantum marginal tube, or a regulator calculation changes the finite shape derivative. Each result matters on its own when it changes a stated hypothesis or conclusion.

Exercise

On one cut, let $\theta_k=-1/L$, $\theta_\ell=-2/L$, and let the entropy terms contribute $+3/(2L)$ and $+1/(2L)$ respectively. Classify the cut and explain why the classical classification differs.

Solution

The quantum expansions are $\Theta_k=+1/(2L)$ and $\Theta_\ell=-3/(2L)$, so the cut is quantum untrapped because the signs are opposite. Geometrically both classical expansions are negative, so it is classically trapped. The difference is physical state dependence in the generalized-entropy derivatives, not a change of null-normal normalization.

References

• Bousso, R., Fisher, Z., Leichenauer, S., and Wall, A. C. (2016). “A Quantum Focussing Conjecture.” Physical Review D 93, 064044. Open PDF.
• Engelhardt, N., and Wall, A. C. (2015). “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime.” JHEP 01, 073. Open PDF.
• Franken, V., Kaya, S., Rondeau, F., Shahbazi-Moghaddam, A., and Tran, P. (2026). “Tests of Restricted Quantum Focusing and a New CFT Bound.” JHEP 05, 111. DOI. Open PDF.
• Wall, A. C. (2013). “The Generalized Second Law Implies a Quantum Singularity Theorem.” Classical and Quantum Gravity 30, 165003. DOI. Open PDF.