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Proof of a Quantum Bousso Bound

Raphael Bousso, Horacio Casini, Zachary Fisher, Juan Maldacena

TL;DR

This work proves the generalized Covariant Entropy Bound for free quantum fields in the regime of weak gravitational backreaction by rigorously defining a regulated, vacuum-subtracted entropy ΔS on null light-sheets. The authors derive a modular Hamiltonian on the light-sheet and show that the bound ΔS ≤ ΔA/(4Għ) follows from two key steps: ΔS ≤ ΔK, via relative entropy, and ΔK ≤ ΔA/(4Għ), via the explicit form of K_L and focusing dynamics encoded in the Raychaudhuri equation. The result does not assume the null energy condition and clarifies the role of ultralocal horizon structure and conformal symmetry in establishing a light-sheet-specific entropy bound. The work also discusses connections to Bekenstein bounds, the generalized second law, and potential extensions to interacting theories and larger backreactions, underscoring the interplay between quantum information, field theory, and gravity.

Abstract

We prove the generalized Covariant Entropy Bound, $ΔS\leq (A-A')/4G\hbar$, for light-sheets with initial area $A$ and final area $A'$. The entropy $ΔS$ is defined as a difference of von Neumann entropies of an arbitrary state and the vacuum, with both states restricted to the light-sheet under consideration. The proof applies to free fields, in the limit where gravitational backreaction is small. We do not assume the null energy condition. In regions where it is violated, we find that the bound is protected by the defining property of light-sheets: that their null generators are nowhere expanding.

Proof of a Quantum Bousso Bound

TL;DR

This work proves the generalized Covariant Entropy Bound for free quantum fields in the regime of weak gravitational backreaction by rigorously defining a regulated, vacuum-subtracted entropy ΔS on null light-sheets. The authors derive a modular Hamiltonian on the light-sheet and show that the bound ΔS ≤ ΔA/(4Għ) follows from two key steps: ΔS ≤ ΔK, via relative entropy, and ΔK ≤ ΔA/(4Għ), via the explicit form of K_L and focusing dynamics encoded in the Raychaudhuri equation. The result does not assume the null energy condition and clarifies the role of ultralocal horizon structure and conformal symmetry in establishing a light-sheet-specific entropy bound. The work also discusses connections to Bekenstein bounds, the generalized second law, and potential extensions to interacting theories and larger backreactions, underscoring the interplay between quantum information, field theory, and gravity.

Abstract

We prove the generalized Covariant Entropy Bound, , for light-sheets with initial area and final area . The entropy is defined as a difference of von Neumann entropies of an arbitrary state and the vacuum, with both states restricted to the light-sheet under consideration. The proof applies to free fields, in the limit where gravitational backreaction is small. We do not assume the null energy condition. In regions where it is violated, we find that the bound is protected by the defining property of light-sheets: that their null generators are nowhere expanding.

Paper Structure

This paper contains 13 sections, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Regulated Entropy $\Delta S$
  4. Proof that $\Delta S\leq \Delta K$
  5. Proof that $\Delta K\leq \Delta A/4G\hbar$
  6. Ultralocality and Conformal Symmetry Determine $\Delta K$
  7. Focusing and Nonexpansion Bound $\Delta A$
  8. Discussion
  9. Relation to other work
  10. Extensions
  11. Monotonicity of $\frac{\Delta A(c,b)}{4G\hbar}- \Delta S$
  12. Free scalar field
  13. Sufficient Conditions For Monotonicity

Figures (3)

  • Figure 1: The light-sheet $L$ is a subset of the light-front $x^-=0$, consisting of points with $b(x_\perp ) \leq x^+ \leq c(x_\perp )$ (a). The light-sheet can be viewed as the disjoint union of small transverse neighborhoods of its null generators (b).
  • Figure 2: Operator algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a spacelike interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself.
  • Figure 3: A possible approach to defining the entropy on a light-sheet beyond the weak-gravity limit. One divides the light-sheet into pieces which are small compared to the affine distance over which the area changes by a factor of order unity. The entropy is defined as the sum of the differential entropies on each segment.