Entanglement Islands from Hilbert Space Reduction

In this paper we propose a mechanism to generate entanglement islands in quantum systems from a purely quantum information perspective. More explicitly we show that, if we impose certain constraints on a quantum system by projecting out certain states in the Hilbert space, it is possible that for all the states remaining in the reduced Hilbert space, there exits subsets $I_a$ whose states are encoded in the states of another subset ${\mathcal{R}}_a$. Then the subsets $\{I_a\}$ are just the entanglement islands of the corresponding subsets $\{{\mathcal{R}}_a\}$. We call such a system self-encoded, and find that the entanglement entropy in such systems should be calculated by a new island formula. We give a comparison between our new island formula and island formula in gravitational theories. Inspired by our mechanism, we propose a simulation of the AdS/BCFT correspondence and the island phases in this context via a holographic CFT${}_2$ with a special Weyl transformation.