Entanglement Islands from Hilbert Space Reduction
In this paper we try to understand the \textit{Island formula} from a purely quantum information perspective. We propose that the island phase is a property of the quantum state and the Hilbert space where the state is embedded in. More explicitly, when certain constraints are imposed in a quantum system by projecting out certain states in the Hilbert space, such that for all the states remaining in the reduced Hilbert space, there exists a mapping from the state of the subset $\mathcal{R}$ to the state of another subset $I$, which we call a coding relation. We call such a system \textit{self-encoded}. In a self-encoded system the way we compute the reduced density matrix changes essentially, which results in a new \textit{island formula} to calculate the entanglement entropy. Furthermore, we propose that, the \textit{island formula} in gravitational theories, which is proposed to rescue the unitarity in the process of black hole evaporation, should be a special application of this new \textit{island formula}. The combination of these two \textit{island formulas} indicates that the effective theory of a gravity is a self-encoded theory with the Hilbert spaces vastly reduced.