The fine-grained entropy of ordinary systems can be computed immediately by describing the density matrix of the associated system. It is not the case for black holes because we need a quantum formulation of this gravitational system. We outline that the von Neumann entropy can be written in terms of classical surfaces on one side, such as RT, HRT, and maximin surfaces, and quantum surfaces on another, such as quantum extremal surfaces and quantum maximin surfaces that extremize or maximize, respectively, the generalized entropy, i.e., \(Area/4+S_{outside}\) . Then, we initiate a total entropy of radiation by combining two blocks, one for the island region behind the horizon and the one for radiation region outside the black hole providing, therefore, a dual perspective, internal and external in contrast to gravitational fine-grained entropy. We suggest that gravitational fine-grained entropy can be written by a new kind of surface called maximinext surface. We provide insights about the new version of island entropy in AdS/CFT framework.