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We study four coupled SYK models and nearly AdS$_2$ gravities. In the SYKmodel side, we construct a model that couples two copies of two coupled SYKmodels. In nearly AdS$_2$ gravity side, we entangle matter fields in two copiesof traversable wormholes. In both cases, the systems show first order phasetransitions at zero temperature by changing couplings, which is understood asthe exchange of traversable wormhole configurations. In nearly AdS$_2$ gravitycases, by exchanging the role of space and time the wormholes are interpretedas bra-ket wormholes. In Lorentzian signature, these bra-ket wormholes lead totwo closed universes that are entangled with each other as well as matterfields in the flat space without dynamical gravity. We study the effect ofprojection or entangling operation for matters on flat spaces and they causephase transitions in bra-ket wormholes, which leads to the pair annihilation ofclosed universes. Using these bra-ket wormholes, we discuss the way to embedstates in 2d holographic CFTs into Hilbert space of many 2d free fields.

Four coupled SYK models and Nearly AdS$_2$ gravities: Phase Transitions in Traversable wormholes and in Bra-ket wormholes