Bekenstein bound in the bulk and AdS/CFT

TL;DR

This paper explores the relationship between bulk and boundary observables in AdS/CFT, introducing the sphere of ignorance concept and deriving a Bekenstein bound in the boundary theory for spherically symmetric excitations.

Contribution

It identifies the boundary modular Hamiltonian with bulk observables for symmetric excitations and introduces the sphere of ignorance, linking bulk entropy bounds to boundary entanglement properties.

Findings

Bound on bulk entropy via boundary relative entropy positivity.

Comparison of Bekenstein and Araki-Lieb bounds in AdS/CFT.

Clarification of differences between pure and thermal states.

Abstract

In this paper, we identify the change in the boundary full modular hamiltonian with the bulk observables for spherically symmetric excitations. The identification is demonstrated for perturbative as well as non perturbative excitations. We introduce the notion of the sphere of ignorance, that describes the bulk region that can not be probed by boundary regions below a certain size. It is argued that the vacuum subtracted entropy in the bulk associated with the sphere of ignorance is bounded by the difference of the change of entanglement entropies for complementary regions in the boundary for spherically symmetric state. Bekenstein bound for the sphere of ignorance reflects itself in the boundary theory as the positivity and monotonicity of the relative entropy of the complementary boundary balls. We compare the proposed bound with Araki-Lieb bound and identify the non-trivial domains where Bekenstein limit sets the lower bound. Moreover, we clarify throughout the paper fundamental differences between pure state and thermal excitations from an information theoretic point.