Quantum Bousso Bound Overview

• Quantum Bousso Bound is a quantum extension of the covariant entropy bound, integrating holography with quantum corrections to constrain entropy flux through light-sheets.
• It leverages quantum information theory by relating area differences and modular Hamiltonian differences to vacuum-subtracted von Neumann entropy.
• Implications include advances in black hole physics, holographic entanglement, and quantum singularity theorems, shaping modern quantum gravity research.

The Quantum Bousso Bound is a quantum extension of the covariant entropy bound, synthesizing elements from holography, quantum field theory, and gravitational thermodynamics. It constrains the fine-grained (state-dependent) entropy flux passing through light-sheets in terms of area differences and quantum corrections, reflecting deep connections between geometric properties of spacetime and the flow of quantum information.

1. Historical Origins and Formulation

The classical covariant entropy bound posits that the entropy $S$ passing through a null hypersurface (light-sheet) from a spacelike surface $B$ is restricted by the initial and final areas as

$S \leq \frac{A[B] - A[B']}{4G\hbar}.$

This conjecture is central in holographic entropy bounds and black hole physics. The Quantum Bousso Bound refines this principle to incorporate contributions from quantum entanglement and allows for settings where classical energy conditions are violated, as in evaporating black holes or quantum cosmological scenarios. The quantum formulation relates geometric area loss directly to regulated (vacuum-subtracted) von Neumann entropy and, in many constructions, modular Hamiltonians or conditional entropies.

2. Mathematical Framework and Quantum Corrections

For free fields and in the regime of weak gravitational backreaction, the entropy on a light-sheet $L$ is defined as a vacuum-subtracted quantity,

$$\Delta S = S(\rho_L) - S(\sigma_L) = -{\rm Tr}(\rho_L \log \rho_L) + {\rm Tr}(\sigma_L \log \sigma_L),$$

where $\rho_L$ is the state restricted to the light-sheet and $\sigma_L$ the vacuum restriction (Bousso et al., 2014).

The Quantum Bousso Bound is then

$\Delta S \leq \frac{A - A'}{4G\hbar},$

with $A$ and $A'$ the initial and final area cross-sections of the light-sheet. The proof leverages quantum information theory: the positivity of relative entropy implies $\Delta S \leq \Delta K$ (where $\Delta K$ is the modular Hamiltonian difference), and the explicit construction of the modular Hamiltonian for null segments yields the full bound via linearized gravitational focusing.

In more general, interacting quantum field theories ($d > 2$), the vacuum-subtracted entropy on a null segment is given by a weighted integral of the null component of the stress tensor (Bousso et al., 2014): $$\Delta S = \frac{2\pi}{\hbar} \int d^{d-2}y \int_0^1 dx^+ \, g(x^+) \langle T_{++}(x^+, y) \rangle,$$ where $g(x^+)$ is theory-dependent but constrained by monotonicity and relative entropy bounds.

3. Quantum Focusing and Energy Conditions

The Quantum Focusing Conjecture (QFC) generalizes classical spacetime focusing to quantum settings by postulating that the quantum expansion,

$$\Theta[V(y); y_1] = \lim_{\mathcal{A}\to 0} \frac{4G\hbar}{\mathcal{A}} \frac{\delta S_{\rm gen}}{\delta \epsilon}|_{y_1},$$

cannot increase under null deformations. The generalized entropy combines geometry and quantum field information: $S_{\rm gen} = \frac{A}{4G\hbar} + S_{\rm out}.$ Integration of the QFC recovers the Quantum Bousso Bound directly (Bousso et al., 2015).

A direct implication of the QFC is the Quantum Null Energy Condition (QNEC),

$$\langle T_{kk} \rangle \geq \frac{\hbar}{2\pi\mathcal{A}} S_{\rm out}''.$$

This lower bound on local energy density in terms of the second variation of the von Neumann entropy has been proven in free and superrenormalizable bosonic field theories (Bousso et al., 2015).

4. Extensions: Higher Dimensions, Boundary Limits, and Generalized Entropy

In higher dimensions, quantum corrections such as buoyancy (from Unruh radiation pressure) become more pronounced. The neutral floating point at which an object can be dropped into a black hole lies extremely far from the horizon unless the object's size far exceeds its Compton wavelength, with the scaling $b/b_C > F(D)$ and $F(D) \sim D^{D/2}$ for large $D$ (Hod, 2011). This suggests that the Quantum Bousso Bound in $D > 3$ dimensions must be supplemented by explicit quantum correction terms, and naive extrapolation from three-dimensional results is unreliable.

At null infinity ($\mathcal{I}^+$) of asymptotically flat spacetimes, boundary versions of the Quantum Bousso Bound, QNEC, and generalized second law have been derived. For example, the boundary QBB is

$\hat{S}_C \leq \frac{2\pi}{\hbar} \Delta \hat{K},$

with $\hat{K}$ obtained via a weighted integral of energy flux and shear over null generators, regulating both matter and geometric contributions (Bousso, 2016).

In gravitational theories with higher derivatives, the classical Bousso bound can be violated unless the area is replaced by Wald entropy (with Iyer–Wald–Wall dynamical corrections). The modified bound reads

$$S_L \leq \frac{1}{4\hbar G_N}[S_{\rm IWW}(B) - S_{\rm IWW}(B')],$$

where $S_{\rm IWW}$ incorporates all curvature and nonlocal dynamical terms (Bhattacharyya et al., 25 Mar 2024).

5. Singularities, Entropy, and Quantum Information

Recent singularity theorems have been established assuming the Quantum Bousso Bound. A spatial region $B$ is "hyperentropic" if $S(B) > A(\partial B)/(4G)$; if entering null geodesics contract, geodesic incompleteness—i.e., a singularity—results (Bousso et al., 2022). This replaces the noncompactness required in Penrose's classical theorem with an informational criterion, tying singular behavior in spacetime directly to excess entropy and quantum information.

Strong entanglement ("hyperentanglement") between regions $B$ and $R$—quantified by a drop in the generalized entropy $S_{\rm gen}(B \cup R) < S_{\rm gen}(R)$—and negative quantum expansion guarantee quantum singularities, which are obstructions to continuing semiclassical evolution even in classically regular spacetimes (Bousso et al., 2022).

Cutoff procedures in cosmological Bousso bounds can be phrased in terms of entropy density versus energy density rather than curvature singularities, leading to geodesic incompleteness from "excess entropy" rather than geometric divergence (Kanai et al., 2023).

6. Discrete Quantum Focusing and Conditional Entropies

A recent reformulation uses a discrete version of quantum focusing based on conditional smooth max-entropy rather than a continuous quantum expansion (Bousso et al., 23 Oct 2024). The discrete max-QFC replaces numerical expansion with a qualitative nonexpansion condition: the conditional generalized max-entropy does not increase under suitable outward null deformations,

$(b|a) \leq 0$

for all accessible wedges $b \supset a$. This axiom is sufficient to derive the Quantum Bousso Bound, QNEC, and strong subadditivity properties of entanglement wedges relevant in holographic dualities, especially where numerical expansions are ill-defined (e.g., at non-smooth boundaries).

7. Applications and Implications

The Quantum Bousso Bound is now foundational in semiclassical gravity, constraining information transfer in black hole evaporation, cosmology, and quantum gravity. It unifies geometric focusing with quantum informational principles:

• Governs entropy flux limits in gravitational radiation (e.g., Hawking radiation, soft theorems).
• Plays a central role in holographic entanglement entropy prescriptions (Ryu–Takayanagi in AdS/CFT, boundary QNEC/GSL).
• Forms the basis for quantum energy conditions, making critical contact with quantum field theory operator structures.
• Arises in the consistency criteria for cosmic censorship and quantum singularity formation, replacing classical trapped surfaces and energy conditions with quantum informational analogues.
• Motivates classical and quantum generalizations under higher-derivative effective field theories via Wald entropy.
• Underlies axiomatic structures in recent discrete entropy formulations, clarifying redundancies and ensuring strong subadditivity in gravitational quantum systems.

The emergence of quantum singularity theorems, discrete focusing axioms, and the interplay between quantum buoyancy and higher-dimensional entropy bounds demonstrates the evolving landscape of quantum holographic constraints, with the Quantum Bousso Bound occupying a central position in the modern understanding of quantum gravity and holography.