Comment on the Bekenstein bound

TL;DR

This paper rigorously derives the Bekenstein entropy bound for finite physical systems using advanced operator algebraic methods, providing a mathematically solid foundation for understanding entropy limits in quantum systems.

Contribution

It introduces a new derivation of the Bekenstein bound employing Connes' bimodules, Tomita-Takesaki theory, and Jones' index within the framework of infinite quantum systems.

Findings

Derivation of the Bekenstein bound using operator algebraic techniques.

Application of von Neumann algebra theory to quantum information limits.

Establishment of a rigorous mathematical foundation for entropy bounds in quantum physics.

Abstract

We propose a rigorous derivation of the Bekenstein upper limit for the entropy/information that can be contained by a physical system in a given finite region of space with given finite energy. The starting point is the observation that the derivation of such a bound provided by Casini [6] is similar to the description of the black hole incremental free energy that had been given by the first named author [23]. The approach here is different but close in the spirit to [6]. Our bound is obtained by operator algebraic methods, in particular Connes' bimodules, Tomita-Takesaki modular theory and Jones' index are essential ingredients inasmuch as the von Neumann algebras in question are typically of type III. We rely on the general mathematical framework, recently set up in [26], concerning quantum information of infinite systems.