Relative entropy and the Bekenstein bound

Abstract: Elaborating on a previous work by Marolf et al, we relate some exact results in quantum field theory and statistical mechanics to the Bekenstein universal bound on entropy. Specifically, we consider the relative entropy between the vacuum and another state, both reduced to a local region. We propose that, with the adequate interpretation, the positivity of the relative entropy in this case constitutes a well defined statement of the bound in flat space. We show that this version arises naturally from the original derivation of the bound from the generalized second law when quantum effects are taken into account. In this formulation the bound holds automatically, and in particular it does not suffer from the proliferation of the species problem. The results suggest that while the bound is relevant at the classical level, it does not introduce new physical constraints semiclassically.

• The paper reinterprets the Bekenstein bound by linking it to the positivity of relative entropy between vacuum and excited states.
• It addresses the species problem, demonstrating that entropy differences remain finite despite the addition of multiple particle species.
• It defines localized entropy and energy by subtracting vacuum contributions, aligning with semiclassical thermodynamic principles.

An Academic Review: Relative Entropy and the Bekenstein Bound

The paper "Relative Entropy and the Bekenstein Bound" by H. Casini critically explores the conceptual and mathematical framework underpinning the Bekenstein bound, enhancing our understanding of entropy limits within the context of quantum field theory (QFT) and statistical mechanics. This research revisits the Bekenstein bound, initially derived from black hole thermodynamics, proposing a fresh interpretation via relative entropy and addressing the proliferation of species problem.

Overview of the Bekenstein Bound

Originally posited in the domain of black hole thermodynamics, the Bekenstein bound suggests a universal upper limit on the entropy $S$ of a bounded physical system in flat space, proportional to its energy $E$ and characteristic size $R$. This is formalized by the inequality $S \leq \lambda ER$, where $\lambda$ is a constant approximately of order one. The bound arises from thought experiments involving the generalized second law (GSL) of thermodynamics in black hole scenarios, which found that the entropy swallowed by a black hole should be offset or exceeded by the change in the black hole's area, as dictated by Einstein's equations.

Key Contributions and Findings

The paper offers significant insights by linking relative entropy and the Bekenstein bound in a novel manner:

• Interpretation via Relative Entropy: It claims that the Bekenstein bound can be effectively interpreted through the positivity of relative entropy between the vacuum state and another state, both restricted to a locale. This makes the statement of the Bekenstein bound robust in quantum flat space by utilizing known results in QFT.
• Resolution of the Species Problem: The proliferation of species problem, which challenges the bound by allowing the entropy to increase indefinitely with particle species, is addressed. The use of relative entropy provides a framework where the difference in entropy between states remains unaffected by an increase in species, thereby protecting the integrity of the bound.
• Localized Entropy and Energy: To make the Bekenstein bound applicable in QFT, localized forms of entropy and energy within a region $V$ are defined. This sidesteps divergences typical in continuous systems by subtracting the vacuum contribution, yielding well-defined, finite quantities for $S_V$ and $K_V$, respectively.
• Semiclassical Implications: The paper argues that these formulations align with the GSL in semiclassical regimes, implying that the boundedness of entropy is a requisite consequence of incorporating quantum effects rather than a novel physical constraint.

Implications and Future Directions

The proposed connection between relative entropy and the Bekenstein bound illuminates the constraints on information storage in physical systems, with profound implications for theoretical physics and information theory. This approach suggests that at both quantum and classical levels, entropy bounds are intrinsic to the fabric of spacetime and physical laws. The elimination of the species problem is particularly compelling, reinforcing the universality of entropy constraints irrespective of the system's complexity.

Looking forward, the insights drawn from this analysis may propel further research into the holographic principle and the nature of quantum gravity. Future work could explore extending these concepts to dynamically evolving spacetimes and rigorously defining these concepts mathematically for arbitrary von Neumann algebras, enhancing our fundamental understanding of information theory in the context of quantum field theories. The research also invites exploration into the implications of these entropy bounds in diverse fields, such as quantum computing, and their potential role in establishing foundational limits on computation and information processing.

Overall, this paper enriches the theoretical discourse on entropy bounds, providing a solid bridge between classical thermodynamics, quantum mechanics, and the ever-perplexing domain of quantum gravity.