  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.
