Quantum Focusing and Generalized Second Laws

The implication is conditional but sharp: if quantum focusing makes the future-directed quantum expansion nonincreasing and a future causal horizon settles so that its expansion tends to zero, then the expansion was nonnegative at every earlier cut, so generalized entropy cannot decrease toward the future. This connects a second-variation conjecture to a generalized second law without making the two statements equivalent.

Required background. Quantum Information supplies von Neumann entropy, relative entropy, and monotonicity, which enter both generalized entropy and independent generalized-second-law proofs.

Helpful background. Quantum Fields in Curved Spacetime supplies state regularity; Einstein Equations supplies backreaction; Black-Hole Geometry and Mechanics supplies causal horizons and null generators; and Semiclassical Gravity fixes the order at which the generalized law is asserted.

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

From focusing to entropy monotonicity

Let $\lambda$ increase to the future along generators of a future causal horizon $H^+$. For a cut $\sigma(\lambda)$ and a fixed outside convention, the quantum expansion obeys

$$\frac{dS_{\rm gen}}{d\lambda} =\frac{1}{4G\hbar}\int_{\sigma(\lambda)}d^{d-2}y\,\sqrt\gamma\,\Theta.$$

For this formula, every generator advances at unit affine speed. More generally a family $V_\lambda(y)$ inserts a nonnegative speed $\partial_\lambda V_\lambda(y)$ in the integrand. Monotonicity of the expansion along such a family requires the full deformation kernel:

$$\frac{d\Theta[V_\lambda;y]}{d\lambda} =\int d^{d-2}y'\, \frac{\delta\Theta[V_\lambda;y]}{\delta V(y')} \frac{\partial V_\lambda(y')}{\partial\lambda}.$$

If the QFC kernel is nonpositive on the allowed future deformations, the nonnegative speeds imply

$$\frac{d\Theta}{d\lambda}\le0,$$

at every generator. A diagonal inequality alone does not justify advancing the whole cut: off-diagonal terms must also be controlled. Now add a future boundary condition: along this family the horizon approaches a stationary state, correlations settle, and $\Theta(\lambda;y)\to0$ as $\lambda\to+\infty$. Because $\Theta$ cannot increase toward the future, every earlier value satisfies $\Theta(\lambda;y)\ge0$. Therefore

$$\frac{dS_{\rm gen}}{d\lambda}\ge0.$$

This is the generalized second law for the selected future causal horizon. The argument and its asymptotic assumption are stated in Bousso et al. 2016, § 6.1, p. 24, PDF. It is an implication of quantum focusing plus late-time behavior, not a proof that arbitrary dynamical or trapping horizons obey the same law.

Why the statements are not equivalent

The generalized second law constrains a first derivative on causal horizons. Quantum focusing constrains second functional derivatives for much more general null deformations. A nonnegative function $\Theta(\lambda)$ can have either sign of derivative, so the GSL alone does not imply QFC. Conversely, QFC without a boundary value allows $\Theta$ to remain negative while decreasing, so focusing alone does not imply the GSL.

There is an independent algebraic route. For a causal horizon with suitable semiclassical expansion, horizon algebra, thermality or boost structure, interactions, and outside-system assumptions, monotonicity of relative entropy yields generalized-entropy monotonicity. Wall states these hypotheses and extensions in Wall 2012, §§ 2.1–2.8 and 3–5, PDF. This proof architecture can establish the GSL in its domain even if an unrestricted QFC is too strong.

The word “horizon” also matters. A future event or causal horizon is defined globally as a boundary of a causal past. An apparent, trapping, or quantum marginal tube is slicing dependent and need not share the algebraic inclusion property used in a causal-horizon proof. Applying the same inequality requires a separate theorem, not a change of vocabulary.

Evidence and limitations

The sources checked through 2026-09-26 support causal-horizon generalized laws under stated assumptions and restricted focusing statements in controlled models. The original pointwise QFC has counterexamples; its unrestricted form is not a general theorem.

Evidence or result	Supported implication	Scope and limitation
Wall 2012, §§ 2.1–2.8, PDF	Relative-entropy monotonicity proves a causal-horizon GSL in a defined semiclassical horizon algebra	Requires the listed horizon, algebra, state, coupling, and renormalization hypotheses
Bousso et al. 2016, §§ 3.3 and 6.1, PDF	QFC plus late-time $\Theta\to0$ implies the GSL on a future causal horizon	QFC is proposed, not proven generally; stationarity is an extra boundary assumption
Bousso et al. 2016, §§ 4–5.3, especially Eqs. (5.8)–(5.15), PDF	Proves off-diagonal focusing from strong subadditivity and reduces the diagonal weak-gravity limit to QNEC, with a free-field proof sketch	QNEC is weaker than the full nonlocal QFC and does not alone prove horizon monotonicity
Franken et al. 2026, § 3, Eqs. (3.13)–(3.20), and § 4, PDF	Proves restricted focusing for specified JT/QFT models and derives a higher-dimensional CFT constraint conditional on rQFC	Published in JHEP 05 (2026) 111; the higher-dimensional implication is not an independent proof of rQFC
Fu, Koeller, and Marolf 2017, §§ 2–3, PDF	Exhibits pointwise QFC violations with a Gauss–Bonnet coupling in $d\ge5$	Specifies a higher-curvature, weak-curvature setting; the pointwise result must be distinguished from proposed smeared EFT statements

In the JT result, the positive total dilaton $\Phi=\phi_0+\phi$ plays the role of transverse area. For one moving endpoint, write $S_{\rm gen}=\Phi/(4G_2)+S_{\rm out}$ plus fixed-endpoint terms, in units $\hbar=1$. Primes denote the chosen null affine derivative, and the corresponding expansions are $\theta=\Phi'/\Phi$ and $\Theta=(\Phi'+4G_2S_{\rm out}')/\Phi$. The field equation and the applicable two-dimensional QNEC give $\Theta'\le-\theta\Theta$, so $\Theta=0$ implies $\Theta'\le0$. This is a restricted zero-expansion statement, not $\Theta'\le0$ for every cut. Their original-QFC counterexamples have matter corrections comparable to the dilaton contribution; they do not establish violations for four-dimensional large black holes with a controlled higher-dimensional reduction.

The competing positions are precise. The conservative position uses the proven causal-horizon GSL where Wall’s algebraic hypotheses hold and treats QFC only as an organizing conjecture. The permissive position expects a suitably restricted focusing principle to explain the same law and its barrier consequences more broadly. A fair comparison fixes state, horizon definition, deformation, regulator, field content, and semiclassical order before asking which statement survives.

Counterexamples and limiting checks

Calculus check. Take $\Theta(\lambda)=Ae^{-\lambda/L}$ with $A,L\gt 0$. It has $d\Theta/d\lambda\lt 0$, tends to zero, and gives increasing generalized entropy. Taking $A\lt 0$ instead approaches zero from below only by increasing, so it violates $d\Theta/d\lambda\le0$. This verifies the direction of the implication.

Boundary-condition counterexample. The constant function $\Theta=-1/L$ satisfies $d\Theta/d\lambda=0$ but violates the GSL. It fails the late-time condition $\Theta\to0$, demonstrating why that hypothesis cannot be omitted.

Horizon counterexample. A slicing-dependent marginal tube can jump between branches while a global causal horizon remains smooth. The causal-algebra nesting used by the GSL proof need not follow the tube, so the theorem cannot simply be transferred.

The open boundary is generic nonequilibrium semiclassical gravity with interacting fields, changing horizon topology, or large metric fluctuations. The conclusions could change after a proof or counterexample settles an appropriately restricted focusing statement in curved interacting QFT, an independent backreaction calculation tests the late-time boundary condition, or a theorem extends relative-entropy monotonicity to a new horizon algebra. Publication of the restricted-QFC calculation does not enlarge its stated domain.

Exercise

Assume $d\Theta/d\lambda\le0$ on a future causal horizon and $\lim_{\lambda\to\infty}\Theta=0$. Prove the sign of $dS_{\rm gen}/d\lambda$, and identify the step that fails if the limiting expansion is $-c\lt 0$.

Solution

Monotonic decrease gives $\Theta(\lambda)\ge\Theta(\infty)=0$ at every finite earlier parameter value. Integrating the nonnegative expansion over the cut gives $dS_{\rm gen}/d\lambda\ge0$. If the limit is $-c$, the same argument yields only $\Theta(\lambda)\ge-c$, which allows negative expansion and therefore does not establish the generalized second law.

References

• Bousso, R., Fisher, Z., Leichenauer, S., and Wall, A. C. (2016). “A Quantum Focussing Conjecture.” Physical Review D 93, 064044. 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.
• Fu, Z., Koeller, J., and Marolf, D. (2017). “Violating the Quantum Focusing Conjecture and Quantum Covariant Entropy Bound in $d\ge5$ Dimensions.” Classical and Quantum Gravity 34, 175006. DOI. Open PDF.
• Wall, A. C. (2012). “A Proof of the Generalized Second Law for Rapidly Changing Fields and Arbitrary Horizon Slices.” Physical Review D 85, 104049. Open PDF.