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We consider pure states in the SYK model. These are given by a simple localcondition on the Majorana fermions, evolved over an interval in Euclidean timeto project on to low energy states. We find that "diagonal" correlators areexactly the same as thermal correlators at leading orders in the large $N$expansion. We also describe "off diagonal" correlators that decay in time, andare given simply in terms of thermal correlators. We also solved the modelnumerically for low values of $N$ and noticed that subsystems become typicallyentangled after an interaction time. In addition, we identified configurationsin two dimensional nearly-$AdS_2$ gravity with similar symmetries. Thesegravity configurations correspond to states with regions behind horizons. Theregion behind the horizon can be made accessible by modifying the Hamiltonianof the boundary theory using the the knowledge of the particular microstate.The set of microstates in the SYK theory with these properties generates thefull Hilbert space.

Pure states in the SYK model and nearly-$AdS_2$ gravity