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Relative entropy and the Bekenstein bound

H. Casini

TL;DR

The authors recast the Bekenstein bound as a fundamental statement about localized quantum information in flat space by defining a finite localized entropy $S_V = S(\rho_V) - S(\rho_V^0)$ and a local energy proxy $K_V$ via the modular Hamiltonian. They demonstrate that the bound $S_V \le K_V$ follows from the positivity of the relative entropy $S(\rho_V \| \rho_V^0)$ and align this with semiclassical proofs of the generalized second law, while addressing the species problem. The framework clarifies how localization-induced divergences cancel and connects to known cases (Rindler wedge, spheres) through explicit modular Hamiltonians. Overall, the work shows that a Bekenstein-type bound naturally arises in quantum field theory without imposing new semiclassical constraints and provides a concrete method for evaluating the bound across diverse settings.

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.

Relative entropy and the Bekenstein bound

TL;DR

The authors recast the Bekenstein bound as a fundamental statement about localized quantum information in flat space by defining a finite localized entropy and a local energy proxy via the modular Hamiltonian. They demonstrate that the bound follows from the positivity of the relative entropy and align this with semiclassical proofs of the generalized second law, while addressing the species problem. The framework clarifies how localization-induced divergences cancel and connects to known cases (Rindler wedge, spheres) through explicit modular Hamiltonians. Overall, the work shows that a Bekenstein-type bound naturally arises in quantum field theory without imposing new semiclassical constraints and provides a concrete method for evaluating the bound across diverse settings.

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.

Paper Structure

This paper contains 6 sections, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A localized entropy
  3. A localized energy
  4. Relative entropy and the Bekenstein bound
  5. Discussion: the Bekenstein bound and the GSL
  6. Appendix: a scalar particle in the Rindler wedge

Figures (1)

  • Figure 1: The horizontal interval represents a spatial sphere (only one spatial dimension shown), while the diamond shaped set is its domain of dependence. The vertical dashed curves are the trajectories of the point transformations induced by the local dynamics in a conformal field theory. The horizontal solid curves are the translation of the sphere for different internal times $s$ (from bottom to top, the curves correspond to integer $s$ from $-5$ to $5$).