Comment on the Bekenstein bound

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 (2008) is similar to the description of the black hole incremental free energy that had been given in Longo (1997). The approach here is different but close in the spirit to Casini (2008). 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 Longo (2018), concerning quantum information of infinite systems.