Bekenstein bound in the bulk and AdS/CFT
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.